Boundary fractal dimension

In a simple technique to determine fractal dimension of a rugged boundary, a series of polygons of side length 1 are constructed on the image of the object. The perimeter of the polygon, P, then becomes the approximation of the perimeter at resolution jS. The boundary fractal dimension, Dl, which varies from... 

In a metered dose inhaler, where fine drug particles are suspended in a propellent, stability and aggregation of the suspension are crucial for the performance of the inhaler. In an investigation into the aerosol formulation stability. Span 80 was added to a suspension of lactose to study the effect on deaggregation under shear.f An increase in shear stress was found to decrease aggregate size and boundary fractal dimension (Fig. 9), which was interpreted as a more compact aggregate. 

Fig. 9 The effect of shear stress on aggregate size and boundary fractal dimension, of 0.08% (w/w) lactose suspended in 1,1,2-trichlorotrifluoroethane in the presence of 0.03% (w/w) surfactant sorbitan monooleate (Span 80). (From Refill)...

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. 

Fig. 12 The variation of apparent permeability coefficient with boundary fractal dimension. (From Ref. " f)...

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.

Image analysis of soil thin sections is the other method that is commonly used for characterizing pore shape. Because of measurement constraints, these analyses are generally conducted in two dimensions, and thus it is the boundary fractal dimension that is used to quantify pore surface roughness (Kampichler Hauser, 1693 Anderson et al., 1996 Pachepsky et al., 1996). The db is defined by the following equation (Mandelbrot, 1983),... 

Allen. M.. G..I. Brown, and N I Miles 1995, Measurement of boundary fractal dimensions review of current techniques Powdei lei linol K-l I 14. 

Allen, M., Brown, G.J., and Miles, N.J. 1995. Measurement of boundary fractal dimensions Review of current techniques. Powder Technol. 84, 1-14. 

Bower et al measured the boundary fractal dimension of lactose aggregates in 1,1,2-trichlorotrifluoroethane using serial perimeter dilation and Adler and Hancock s distance transform function [56]. They found that it decreased as shear rate and aggregate mass fractal dimension increased. 

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. 

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 2.25. Distribution of the boundary fractal dimensions of the carbon black profiles of Figure 2.15(a).

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...

B. H. Kaye, G. G. Clark, Y. Kydar, Strategies for Evaluating Boundary Fractal Dimensions by Computer Aided Image Analysis. Part. Part. Syst. Charact. 11 (1994) 411—417. 

To describe a fractal such as that of F >ue 7.13(b)(iii) one uses what is known as a density of mass fractal. This is a measure of how the system occupies space. This fractal dimension is difierent from the boundary fractal dimension used in Chapter 2 to describe the ru ed boundary of a dense fineparticle. If one grows several a omerates of the W hitten Sander type they may look different but are of the same mass fractal... 

Figure 7.15. The way in which a cluster grows determines the overall ruggedness of the profile and that, in turn can affect the aerodynamic behaviour of the cluster, a) Simulated clusters grown using various probabilities that a wandering subunit will join the cluster, b) Real fumed thorium dioxide a omerates, all having the same aerodynamic diameter, but different boundary fractal dimensions as shown.

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. 

Figure 3 The "coastline" of a Kock Triadic Island is infinite and has a single boundary fractal dimension of 1.26.

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