Scattering from Fractal Objects

The first term comes from the self-correlation of the particle. Discuss the motivation for the second term and relate a to df (the fractal dimension) and d (the dimension of space). What is the structure factor for scattering from this object and discuss it in the limits of small, intermediate, and large wavevectors. 

In this chapter the dilute particulate system, the nonparticulate two-phase system, and the periodic system are discussed in Sections 5.2, 5.3, and 5.5, respectively. Section 5.4 deals with scattering from a fractal object, which may be regarded as a special kind of nonparticulate two-phase system. The soluble blend system is dealt with in Chapter 6. The method discussed in Section 4.2 for determining, for a single component amorphous polymer, the thermal density fluctuation from the intensity I(q) extrapolated to q -> 0 can also be regarded as a small-angle technique. 

In Section 5.2.2 it was shown that at large q the intensity I(q) of scattering from a sphere decays as q A, from a thin disk as q 2, and from a thin rod as q l. The power-law exponent at large q is therefore seen to be related to the dimensionality of the scattering object. There are, however, many cases in which the intensity varies as an unexpected or even fractional power of q. In the case of a Gaussian model of a polymer chain, the intensity was seen to decrease as q 2 even though a chain obviously is a three-dimensional object. The inverse power-law exponents that differ from 1, 2, or 4 can be explained in terms of the concept of a fractal. 

As mentioned in Chap. 1, randomly hyperbranched chains are even more complicated than dendrimers. It has not been completely clear whether they are fractal objects [11, 12] and whether those previously reported M-dependent intrinsic viscosities from an on-line combination of the size exclusion chromatograph (SEC) with viscosity and multi-angle laser light scattering (MALLS) detectors actually captured its structure-property relationship [11, 13]. In the next section, we ll discuss our experimental results in detail. 

In the case of mass fractals, scattering will occur from the bulk of an object instead of surface. As a consequence, when q >... 

From the Eq. (6) it follows, that the minimum value is achieved at and is equal to 2.5, that according to the Eqs. (l)-(3) corresponds to structure fractal dimension J 2.17. Since the greatest value of any fractal dimension for real objects, including at that, cannot exceed 2.95 [8], then from the Eq. (6) the limiting value for the indicated matrix polymer can be evaluated, which is equal to 21.5 mcm. Let us also note that the large scatter of the data in Fig. 14.1 is due to the difficulty of the value precise determination. 

Evolution of silica within the organic matrix was followed by SAXS using a Kratky camera and a linear position-sensitive detector. The structure was described by the size of silica clusters and by fractal dimensions, Dm, characterizing compactness of the object. The fractal dimension was determined from the slope of the linear part of a double logarithmic SAXS plot of the scattered intensity I vs. scattering vector q (=(47T /X)sin 0), where 20 is the scattering angle. 

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