Modern Aspects of Small-Angle Scattering
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These include investigations of nucleic acid-protein complexes; lipoproteins; time-resolved measurements of phospholipid phase transitions; porosity in ceramics; phase separation and defect agglomeration in metals and alloys; catalysts; complex fluids; bulk polymers; and dissolved polymers under flow conditions. The capabilities offered by SAS for exploring such properties as size, shape, structure, morphology, dispersity and interactions of scatterers, on the scale between atomic and macroscopic, are demonstrated.
The volume extends and supplements basic texts. It is intended for new practitioners, scientists active in SAS who wish to learn more about applications outside their immediate expertise, and those desirous of exploring the potential applicability of SAS to their research. Select Parent Grandparent Teacher Kid at heart. Age of the child I gave this to:. Hours of Play:. Tell Us Where You Are:.
Preview Your Review. Thank you. Your review has been submitted and will appear here shortly. Extra Content. Table of Contents Preface. Instrumentation for Small-Angle Scattering; J. Contrast Variation; H. Sequeira, G. Schaefer, R. Brow, B. Olivier, T. Rieker, G. Beaucage, L.
Hrubesh, J. Small-Angle Scattering of Catalysts; H. The algorithm was introduced by Voss in [ 28 ] and defines lacunarity as the entropy of the discreet pixels on the digital image of the fractal. The algorithm begins by the consecutive covering of an image with the grid of nonoverlapping square boxes of the size r , as shown in Figure 2. The total number of boxes in the grid that cover the image is denoted as N. The probability function that a box of size r contains M number of pixels is then defined by.
Thus, the lacunarity can be interpreted as the fluctuations of mass distribution over its mean.
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The lacunarity of the deterministic fractals shows periodicity in the spectrum [ 1 ]. As it will be shown in the next section, some structural properties of deterministic mass fractals can be extracted from such behavior. Although, there are few definitions of lacunarity and several algorithms for its computation exist, we will use here an intuitive and elegant probabilistic approach, which is easily performed computationally [ 28 ].
In spite of this simplicity, it has slight disadvantages in comparison with the gliding-box GB algorithm, which provides more precise and hence more time-consuming evaluations [ 29 ]. The GB algorithm calculates the lacunarity by placing the square box of the size r in the corner of the image of the size L , and then glides the box pixel by pixel along horizontal and vertical directions, note that the box should not slide beyond the image.
When L and r are of the same order of magnitude, the difference in number of boxes for both algorithms is negligible, but when L is at least one order larger than r , the number of boxes will differ in two orders. Both algorithms calculate the number of pixels within the box, thus the computational time directly depends on the number of boxes.
In this section, we present the mathematical description of a very well-known fractal generating algorithm and discuss various types of fractals constructed using deterministic and random approaches. There is no universal method to construct a fractal, but one of the most common algorithms to generate a large class of fractals is iterated function systems IFS [ 19 ]. The IFS image is defined as being the union of geometric transforms of itself.
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Then, in order to obtain the fractal structure, we establish the rule of evolution generator [ 30 , 31 ], shown in Figure 3. Then, translate the obtained circles so that they are situated in the vertices of a square. To generate the structure of the fractal repeat the same rule for each new circle. The fractal that we obtain is a Cantor fractal. The corresponding IFS coefficients of the contraction mappings that generate this fractal are presented in Table 1.
The fractal dimension of Cantor-like fractal is determined by [ 2 ].
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As it can be seen, the value of the fractal dimension depends on how many copies are created at each iteration, in the terms of IFS, the number of the contraction mappings and on the scaling factor. However, the fractal dimension is completely independent on the translations of the copies, and its value can be the same for different textures, as for the models shown in Figure 4 for which the translation coefficients of one of the contraction mappings have different values, presented in Table 2. Note that the transformation coefficients of the fractals presented in Figure 5 are not modified.
Translation coefficients of one of the contraction mappings of the Cantor-like fractals construction.
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We calculate the lacunarity spectra according to Eq. The results are shown in the left part of Figure 6. At first, one can find that the fractal-d has the highest value of the lacunarity along all ranges, which is due to the most inhomogeneous and clustered distribution of the mass among all four models. On the contrary, the texture of the fractal-a has uniformly distributed mass and thus, the lowest lacunarity. The values of the scattering intensities for the fractals -b, -c and -d are scaled up for clarity by the factor 2, 4, and 8, respectively. In addition to differentiating the texture, the lacunarity analysis also may reveal some geometrical and fractal properties.
For example, when one covers the fractal by the boxes of the exact size as the size of its elements at the particular iteration m , the number of empty boxes takes maximum value. This leads to the highest variation in the mass distribution over the mean and the lacunarity at this scale will increase.
The number of such maxima denoted by vertical lines in Figure 6 , left part shows the iteration number of the fractal. The positions of these maxima reveal the size of the units at particular iteration, and from the periodicity of such maxima, one can obtain the scaling factor.
The SAS data, on the other hand, gives information about structure in the reciprocal space. The typical SAS spectrum consists of the region with a constant intensity at small values of q which is called Guinier region. A main feature of the SAS from fractals is that the slope in the region that immediately follows the Guinier regime, so-called fractal region, gives the fractal dimension of the fractal, as discussed in the Introduction section.
The scaling factor of the fractal can be obtained from the periodicity of the minima [ 14 ]. Additionally, the asymptotic behavior of SAS spectrum at high values of q provides the information about number of basic units Nm at particular iteration [ 14 ].
A more general way to construct fractals may be thought in a framework of fat fractals, when the scaling factor is not constant but it depends on the iteration number [ 20 , 32 ]. Here, we present a simple model of the fat fractal, represented by a two-dimensional deterministic Cantor-like mass fractal, as shown in Figure 7. The superscript index … denotes to which iteration number the scaling factor belongs. The fat fractal does not have a unique value of the fractal dimension at every scale, since the scaling factor is not constant. The comparison of the SAS and the lacunarity spectra between thin and fat fractal models is demonstrated in Figure 8.
However, such problem may be addressed using the gliding-box approach [ 29 ]. In the SAS spectrum, the difference between fat and thin fractals may be determined in the fractal region, from the different position of the minima, which correspond to the most common distance between units of the fractal.
In the case of the fat fractal the most common distance is shorter than in the case of thin one, thus in the reciprocal space we observe a minimum corresponding to fat fractal, which is shifted to higher value of q. The behavior of scattering curves of both fat and thin fractals is similar at Guinier and asymptotic regions due to the same overall size and equals the number of units.
One of the most known stochastic algorithms for the construction of the fractals is the Chaos game representation CGR [ 19 ], which is based on the random IFS. The CGR approach allows one to visually reconstruct a great number of the different types of fractals, from well-known deterministic fractals to various classes of disordered systems.
Technically, CGR is an iterative map that generates the position of units, which cover the attractor of IFS, the image of the fractal. CGR algorithm is very convenient for structural investigations using SAS, because it generates directly the coordinates of the scatters, which can be used in the optimized Debye formula [ 25 ]. Here we are interested, how the set of the points generated using the CGR approach will recover the structure of the deterministic fractal.
In order to quantitatively analyze the similarities and the differences in the structure of the fractals obtained by both algorithms, we calculate corresponding SAS and lacunarity spectra. It is seen from Figure 9 that the CGR Cantor fractal approaches the structure of the deterministic Cantor fractal with increasing the number of generated points scattering units k.
To determine the number of generated points in the CGR algorithm needed to obtain the approximation of the deterministic Cantor fractal, we compare the particular iteration, the structure factor, and the lacunarity of the deterministic fractal, and the structure generated from CGR, respectively. The results are shown in Figure The left part of the figure shows almost perfect agreement of the spectra of lacunarity. The positions of the maxima show the sizes of the points in CGR and the sizes of the units at m -th iteration for deterministic fractal.
Note that the size of the points of CGR algorithm is kept constant for any k. In general, the lacunarity has dependence on the sizes of the points, the larger points leading to smaller gaps and to lower lacunarity. The Guinier regions coincide, showing that the overall sizes of the CGR and deterministic fractal are the same.
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The scattering curves almost completely overlap each other in the intermediate region, except the last minimum. The values of the slopes of the curves, which reveal the fractal dimension is approximately the same.