Dendritic Fractality

Determination of the fractal nature of a dendritic surface was first reported by Avnir and FarinJ911 Employing data obtained from solvent accessible surface area analysis (see Sect. 2.3.3.3) for the PAMAM dendrimers, a surface fractal dimension (D) was derived. Two methods were used. The first method applied the relation, 

Similar analyses of Asas data for a much more compacted pentaerythritol-based polyether dendrimcr1851 gave D = 1.96. Comparison of the fractal dimensions for both series, i.e., the PAMAMs and polyethers, suggests structural differences. For D = 1.96, surface 

---

These starburst dendrimers have been subjected 47 to two different fractal analyses I48 49 (a )A c/2 D)/2, where A is the surface area accessible to probe spheres possessing a cross-sectional area, o, and the surface fractal dimension, D, which quantifies the degree of surface irregularity and (b) A = dD, where d is the object size. Both methods give similar results with D = 2.41 0.04 (correlation coefficient = 0.988) and 2.42 0.07 (0.998), respectively. Essentially, the dendrimers at the larger generations are porous structures with a rough surface. For additional information on dendritic fractality, see Section 2.3. 

Fig. 14a-c. Mathematical models of dendritic/fractal systems a) Bifurcation of a line segment with connectivity paths to define Cantor dust, b) Bifurcation of a triangle (two-dimensions) to define Koch snowflakes, c) Fitting of a sphere (three-dimensional) to define a series of infinite nesting spheres... 

Dendritic fractals can very easily be generated via electrolytic reduction of silver ions. 

The growth of crystallization phenomenon sometimes forms nontrivial and beautiful stmctures. Some of the crystals are associated with polyhedral stmcture, but more complex dendrite/fractal and seaweed structures can also appear [105, 106]. Dendrites are anisotropic ramihed structures with nearly paraboloidal growing... 

FIG. 6 Morphology diagram. Plotted is the dimensionless supercooling A versus the crystalline anisotropy e. Compact dendritic (CD) and compact seaweed (CS) structure comprise most of the diagram, under the influence of noise, fractal seaweed (FS) and fractal dendrites (FD) are encountered. For details see text and Ref. 108. 

CONSEQUENCES OF THE FRACTAL CHARACTER OF DENDRITIC HIGH-SPIN MACROMOLECULES ON THEIR PHYSICOCHEMICAL PROPERTIES... 

Fractal Character of Dendritic High-Spin Molecules.40... 

As already mentioned, we chose three different physicochemical properties for studying the influence of the surface area and fractal dimension in the ability of dendritic macromolecules to interact with neighboring solvent molecules. These properties are (a) the differential chromatographic retention of the diastereoisom-ers of 5 (G = 1) and 6 (G = 1), (b) the dependence on the nature of solvents of the equilibrium constant between the two diastereoisomers of 5 (G = 1), and (c) the tumbling process occurring in solution of the two isomers of 5 (G = 1), as observed by electron spin resonance (ESR) spectroscopy. The most relevant results and conclusions obtained with these three different studies are summarized as follows. 

Fractals are mathematically defined self-similar structures (Fig. 1.11) [26]. The scaffold of cascade or dendritic molecules is fractal if the atoms are considered to be points and the bonds to be strictly one-dimensional lines. Self-similarity... 

Rule 1 Dendritic structures consist of self-similar units (fractals). 

This section attempts to examine macromolecular geometry, and in particular dendritic surface characteristics, from the perspectives of self-similarity and surface irregularity, or complexity, which are fundamental properties of basic fractal objects. It is further suggested that analyses of dendritic surface fractality can lead to a greater understanding of molecule/solvent/dendrimer interactions based on analogous examinations of other materials (e.g., porous silica and chemically reactive surfaces such as found in heterogeneous catalysts). 52 ... 

Rabouille, Cortassa, and Aon[81 dried protein, glycoprotein, or polysaccharide containing brine solutions that resulted in dendritic-like fractal patterns. The fractal dimension, D = 1.79, was determined for the pattern afforded by an ovomucin-ovalbumin mixture (0.1 M NaCl). Similar D values were obtained for dried solutions of fetuin, ovalbumin, albumin, and starch the authors subsequently suggest that fractal patterning is characteristic of biological polymers. 

In 1990, it had been proposed1851 that the repetitive, generational, branching topology inherent in dendritic structures could be characterized as being fractal. Support for this conjecture was provided by a computational1861 examination of the solvent accessible surface area (ASas) of dendrimers at different generations. 

Figure 2.14. Employing the log of experimentally determined dendritic diameters the fractal dimension D — 2.29 was found for a series of acid terminated dendrimers. [See the table presented in Figure 2.8]...

Avnir et al. llbl have examined the classical definitions and terminology of chirality and subsequently determined that they are too restrictive to describe complex objects such as large random supermolecular structures and spiral diffusion-limited aggregates (DLAs). Architecturally, these structures resemble chiral (and fractal) dendrimers therefore, new insights into chiral concepts and nomenclature are introduced that have a direct bearing on the nature of dendritic macromolecular assemblies, for example, continuous chirality measure44 and virtual enantiomers. ... 

---