Fractals in Solution

Rice and Lin (1994) measured fractal geometry by small-angle x-ray scattering. In solid state, the fractal dimensions were between 2 and 3 which confirms the visual observations of Krasner et al (1996) (see section 2.5.4). While in the solid state the organics were surface fractals, in solution they are expected to become mass fractals. Osterberg and Mortensen (1994) measured fractal structure using small angle... 

Ren, S.-Z., Tombacz, E. and Rice, J.A. (1996). Dynamic light scattering from fractals in solution Application of dynamic scaling theory to humic acid. Phys. Rev. E, S3, 2980-2983. 

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. 

Relevant Facts. Microscopic examination of the surface of ionically doped polypyrrol shows a fractal surface. The real surface area can be probed by using organic compounds (e.g., jMiitrophenol) of various sizes and finding, by UV-visible measurements of the change in solution concentration caused... 

Because each monomer is potentially tetrafunctional, all sorts of partially hydrolyzed and partially condensed species are possible. For example, 10 distinct dimers may be formed. Many cyclic species can occur. Many of the species in solution are metastable with respect to anhydrous, amorphous Si02, and rearrangements occur with time (2). The complexity of this system means that the structure and growth of these polymers can be described only in a statistical and geometric fashion rather than by a more conventional description in terms of topologies (such as chains and cross-links) and molecular weights. The most useful and intuitive description of the structure of these polymers is in terms of fractal geometry. 

However, polymer coils overlap and dominate most of the physical properties of semidilute solutions (such as viscosity). Thus, adding a very small amount of polymer to a solvent can create a liquid with drastically different properties than the solvent. This unique feature of polymer overlap is due to their open conformations. Linear polymers in solution are fractals with fractal dimension I) < 3. In semidilute solutions, both solvent and other chains are found in the pervaded volume of a given coil. The overlap parameter P is the average number of chains in a pervaded volume that is randomly placed in the solution ... 

Techniques have been developed to determine the fractal dimension of experimental aggregate particles in solution using small-angle scattering techniques, from x-ray, light, or even neutron sources. In these techniques the scattering intensity, I q), is proportional to the scattering vector, q, raised to the mass fractal dimension by ... 

This section analyzes the scaling properties of the uptake curve M t)/Moo on/across fractals in a single theoretical framework. The (fractional) uptake curve M t)/Moo is the ratio of the solute quantity entering the structure up to time t and the quantity entering at saturation (i.e. at t —> oo), i.e. 

Rapid Method of Estimating the Fractal Dimension of Macromolecular Coils of Biopolymers in Solution... 

In this chapter a method is proposed for finding the fractal dimension (D) of a macromolecular coil in solution, which uses only the characteristic viscosity of the polymer in an arbitrary solvent and a 0 solvent. Several examples are given to illustrate the applicahility of the proposed method to biopolymers of different classes. From D, one can determine a number of other important parameters characterising the behaviour of polymers in dilute solutions. 

Linear polymer macromolecules are known to occur in various conformational and/or phase states, depending on their molecular weight, the quality of the solvent, temperature, concentration, and other factors [1]. The most trivial of these states are a random coil in an ideal (0) solvent, an impermeable coil in a good solvent, and a permeable coil. In each of these states, a macromolecular coil in solution is a fractal, i.e., a self-similar object described by the so-called fractal (Hausdorff) dimension D, which is generally unequal to its topological dimension df. The fractal dimension D of a macromolecular coil characterises the spatial distribution of its constituent elements [2]. 

All these methods require quite complex and laborious measurements [3-5]. The simplest of these methods, which requires no sophisticated instrumentation, is measurement of [q], which can be performed in virtually any laboratory. Therefore, in this work, we propose a simple rapid method of estimating the fractal dimension (D) of macromolecular coils in solutions, which is based on the same principles as applied in deriving Equations (16.4)-(16.6). The coefficient of swelling of a macromolecular coil is known to be defined as [6] ... 

---