Structural boundary fractal dimension

Fig. 3 Fractal dimensions can be used to evaluate the rugged structure of fine particles. (A) Fractal dimensions used to describe the ruggedness of various lines (B) physical basis of the equipaced exploration technique for evaluating the fractal dimensions of rugged boundaries (C) data generated by the equipaced exploration technique for the profile of (B) c5s, structural boundary fractal dimension c5x, textural boundary fractal dimension.

Figure 2.24. Some profiles contain regions which display obviously different fractal structure. These variations are lost if the profile is treated as a whole. In order to examine the varying fractal structure, the r ons can be isolated and used to create synthetic islands which can then be characterized to extract more detailed information, a) The coasdine of Great Britain examined as a whole, b) A synthetic island generated by joining two copies of the west coast displays a significandy larger structural boundary fractal dimension than the data for the whole profile, c) A synthetic island created from the east coast of Great Britain displays a lower structural boundary fractal than the island as a whole.

Figure 2.26. Summary of variations in the structural boundary fractal of two populations of carbonblack profiles [38]. a) Tracings of profiles of two carbonblack populations produced by different methods, b) Distribution functions of the structural boundary fractal dimensions for...

Fig. 10 Schematic diagram showing that (A) for a compact aggregate, the particles on the edge will be shed first because of the lower interparticle forces because of less nearest neighboring particles. The final aggregate will have a lower boundary fractal dimension, although the structure compactness is preserved. (B) For a loose aggregate, after the rupture of the chain structure, some aggregates (shown in gray color) are more compact and also have lower boundary fractal dimension.

Fractal dimension was applied to characterize the internal structure of porous films made from ethylcel-lulose (EC) and diethylphthalate (DEP). Drug permeation was found to correlate with boundary fractal dimension on a semilog plot (Fig. 12). " However, Dl simply describes the ruggedness of a line and does not represent the porosity. More work is required for fractal dimensions in this case. 

It is interesting to compare eq. (15) with the results obtained on finitely ramified fractals by means of Green function renormalization [9-10]. It has been shown that the fractional uptake curve for a structure possessing fractal dimension dj, walk dimension d, and adsorbing from a reservoir at constant concentration c through an exchange manifold B (which represents the permeable boundary for treuisfer) possessing fractal dimension d scales as... 

In all the cases examined so far, it is the matter distribution of the object that has exhibited the property of self-similarity. These objects are called mass fractals. Other situations are encountered, where it is not the matter distribution which has self-similarity, but rather the pore distribution in these cases, we speak of pore volume fractals. Some structures are found in which only the contour or surface manifests scale invariance these are called boundary or surface fractals, and the exponent we need to know is the boundary fractal dimension. To obtain the corresponding exponents, we calculate the autocorrelation function, the mass distribution or the number of boxes, restricting ourselves to the relevant subsets (the points occupied by matter, the points in the pore volume, or the points lying in the interface). 

Quantitatively, a self-afRne fractal is defined by the fact that a change Ax XAx (and possibly Ay —> XAy) transforms Az into X Az, where H lies between 0 and 1. The case H — 1 corresponds to a self-similar fractal. Self-affine fractal structures are no longer characterised by just one (mass or boundary) fractal dimension they require two. The first is local and can be determined by the box-counting method, for example it describes the local scale invariance and its value lies between 1 and 2. The second is global and its value is a simple whole number describing the asymptotic behaviour of the fractal. In the case of a mountain, this global dimension is simply 2. When viewed from a satellite, even the Himalayas blend into the surface of the Earth. 

It can be seen that, particularly for the denser structures, one can also draw a boundary fractal around the shape and that, as the mass fractal dimension gpes up, the boundary fractal dimension goes down. At 1 % probability of joining, the a omerate grown is very dense. In the study of the colloidal precipitates one can have structures anywhere intermediate in the series shown in Figure 7.15. As demonstrated by the profiles of a set of thorium dioxide fumes, the boundary fractal dimension of profiles that all have the same aerodynamic diameter vary dramatically as shown in Figure 7.15(b). Measured boundary fractal dimensions of these profiles are shown under the profiles. 

In the evolution of solids from solution, a wide spectrum of structures can be formed. In Fig. 4, a simple schematic representation of the structural boundary condition for gel formation is presented. At one extreme of the conditions, linear or nearly linear polymeric networks are formed. For these systems, the functionality of polymerization /, is nearly 2. This means there is little branching or cross-linking. The degree of cross-linking is nearly 0. In silica, gels of this type can be readily formed by catalysis with HCl or HNO3 under conditions of low water content (less than 4 mol water to 1 mol silicon alkoxide). The ideal fractal dimension for such a linear chain structure is 1. The phe-... 

The use of fractal analysis makes it possible to relate molecular parameters to characteristics of supermolecular structure of polymers. Figure 11.12 illustrates the linear correlation between D and df [dj was estimated from Equation (11.27)] for epoxy polymers. When the molecular mobility is suppressed (D = 1), the structure of the polymer has the fractal dimension df = 2.5, which corresponds to p. = 0.25. The given value of the Poisson coefficient corresponds to the boundary of ideally brittle structure at p< 0.25, the polymer is collapsed without viscoelastic or plastic dissipation of energy [3]. This is fnlly consistent with the Kansch conclnsion [117] stating that any increase in the molecular mobility enhances dissipation of the mechanical energy supplied from the outside and, as a conseqnence, increases plasticity of the polymer. When D = 2 the df value is equal to 3, which corresponds to p = 0.5, typical of the rubbery state. 

Environmental particles, microbial colonies, and even patterns of movement of organisms can be characterized in terms of several different fractal dimensions [14] (Table 1.1). The fractal dimension of the surface Ds (boundary/interface) of a solid structure is obviously an important characteristic, but many of the physical properties of solids also depend on the scaling behavior of the entire solid and/or of its pores. Systems where surface and mass scale similarly are termed mass fractal systems, those where surface and the pore volumes scale similarly are described as pore fractals and systems where only the surface is fractal are designated as surface fractals (Table 1.1 Figure 1.2). 

Thus, the simple method of estimation of the surface fractal dimension d of nanoclusters for the structure of crosslinked epoxy polymers, which are considered as natural nanocomposites, was offered. The lower boundary of 2.55 indicates that packing of nanoclusters is less dense in comparison with an ideal one, for which surf expected. Unlike inorganic nanofiller nanoparticles, nanoclusters in... 

As it is known, autohesion strength (coupling of the identical material surfaces) depends on interactions between some groups of polymers and treats usually in purely chemical terms on a qualitative level [1, 2], In addition, the structure of neither polymer in volume nor its elements (for the example, macromolecular coil) is taken into consideration. The authors [3] showed that shear strength of autohesive joint depended on macromolecular coils contacts number A on the boundary of division polymer-polymer. This means, that value is defined by the macromolecular coil structure, which can be described within the frameworks of fiactal analysis with the help of three dimensions fractal (Hausdorff) spectral (fraction) J and the dimension of Euclidean space d, in which ifactal is considered [4]. As it is known [5], the dimension characterizes macromolecular coil connectivity degree and varies from 1.0 for linear chain up to 1.33 for very branched macromolecules. In connection with this the question arises, how the value influences on autohesive joint strength x or, in other words, what polymers are more preferable for the indicated joint formation - linear or branched ones. The purpose of the present communication is theoretical investigation of this elfect within the frameworks of fractal analysis. 

---