Fragment self-similarity

In a similar context, fragment self-similarities have been used to derive QSAR models and identified active molecular sites in molecules. 

This paper is divided into two main, interconnected parts—breakup and coalescence of immiscible fluids, and aggregation and fragmentation of solids in viscous liquids—preceded by a brief introduction to mixing, this being focused primarily on stretching and self-similarity. 

Size distributions of fragments produced in a single rupture event are self-similar... 

Conductivity. Let a(/j be the conductivity of a fragment of the fractal structure of dimension /, where Iq < l < E,. Because of the self-similarity of the structure, the ratio of conductivities on different scales l and / is defined only by the ratio of the scales ... 

FIG. 2 Evidence for self-similarity of the aggregate mass distribution for the fragmenting system (aluminum oxide/polyacrylic acid) according to Eq. 9. 

The notions of functional self-similarity and scaling arise very naturally in polymer science and have found applications in this field for many years. Thus, a linear polymer chain with excluded-volume interactions is a perfect example of a physical object to which scaling should be applicable, as a subdivision of the entire chain into a collection of blobs or Kuhn macrosegments [9,14]. One of these segments may be envisioned as a fragment of sufficient length to ensure that its statistical properties are effectively independent from the remainder of the chain. 

Thus, the averaged square end-to-end separation of the iV-bond chain is related to that of a smaller X-bond fragment by the proportionality factor N/K. Bond correlations due to nonlocal (excluded-volume) interactions complicate the problem, but it is still reasonable, as we have seen, to assume that subunits of the system will, in some sense, be replicas of the entire chain. More generally, the many examples presented in the preceding sections illustrate that there are many properties of physically interesting systems that can be analyzed successfully from this point of view and that, in particular, the appropriate application of the assumption of self-similarity can be used to generate reliable approximations for physical quantities that we otherwise would be unable to calculate. 

Typical examples of these fractals are the Cantor set ( dust ), the Koch curve, the Sierpinski gasket, the Vicsek snowflake, etc. Two properties of deterministic fractals are most important, namely, the possibility of exact calculation of the fractal dimension and the infinite range of self-similarity -°° +°°). Since a line, a plane, or a volume can be divided into an infinite number of fragments in different ways, it is possible to construct an infinite number of deterministic fractals with different fractal dimensions. Therefore, deterministic fractals cannot be classified without introducing other parameters, apart from the fractal dimension. 

With the assumption that = const and K = 1.0, = 0.85 was obtained with L = const and K > 1.0, the calculations gave % = (0.85) = 0.926. The calculation of the dimension D from Equation (11.55) gave 1.17. Thus, when the condition mentioned above is fulfilled, the section of a macromolecule between chemical crosslinking points can be represented as a fractal. Variation of K changes the crosslinking density and, hence, L. As a consequence, the chain ceases to be a self-similar fractal. Nevertheless, it can still be modelled by a non-homogeneous fractal with the same fragmentation step but with a variable D value, determined from relationship (11.39). The D values calculated in this way are listed in Table 11.6. 

Molecular Basis of LEER. Modeling of the Electronic Substituent Effect Using Fragment Quantum Self-Similarity Measures. 

The term fractal and the concept of fractal dimension were introduced by Mandelbrot [1]. Since Mandelbrot s work, many scientists have used fractal geometry as a means of quantifying natural structures and as an aid in understanding physical processes occurring within these structures. Fractals are objects that appear to be scale invariant. Mandelbrot defines them as shapes whose roughness and fragmentation neither tend to vanish, nor fluctuate up and down, but remain essentially unchanged as one zooms in continually and examination is refined . The above property is called scale invariance . If the transformations are independent of direction, then the fractal is self-similar if they are different in different directions, then the fractal is self-afflne (see Chapter 2). 

Vigil and ZilF (1989) dispense with the assumption of binary breakage in their analysis of self-similarity but appear to assume a constant mean number of fragments independently of the size of the fragmenting particle. 

The mathematical statement of the inverse problem is as follows Given measurements of F x, t), the cumulative volume (or mass) fraction of particles of volume ( x) at various times, determines, b x), the breakage frequency of particles of volume x, and G x x ), the cumulative volume fraction of fragments with volume ( x) from the breakage of a parent particle of volume x. Obviously, the experimental data on F x, t) would be discrete in nature. We assume that G x x ) is of the form (5.2.9) and rely on the development in Section 5.2.1.1 using the similarity variable z = b x)t. Self-similarity is expressed by the equation F x, t) = 0(z), which, when substituted into (6.1.1), yields the equation... 

In what follows, we let u represent the ratio of the breakage rate of the fragment to that of the parent particle. The statement of the inverse problem lies in calculating the unknown function g u) over the unit interval and the constant P given the self-similar curve in the form of 6 versus z. Since g u)... 

A quantum-chemical study of the mechanism of the condensation reaction between propanoic acid and aniline has been made. The mechanism of proton-transfer reactions of 3,5-dinitrosalicylic acid (41) has been the subject of a review (50 references). A successful correlation has been obtained between dissociation of a variety of acids in different solvents and quantum self-similarity measures of the CO2H fragment. The substituent effects of the isopropyl group in 2- (42), 3- (43), and 4-isopropylbenzoic acid (44) have been evaluated from their enthalpies of formation, gas-phase acidities, acidities in MeOH and in DMSO, and their IR spectra in tetrachloromethane. Particular attention was given to the influence of variable conformation on the observed steric effect. In contrast to 2-r-butylbenzoic acid and similarly to 2-methylbenzoic acid, 2-isopropylbenzoic acid exists in two planar conformations (42a,b) in equilibrium. Owing to this conformational freedom, the... 

---