Polymer fractal dimension

Based on the fractal behavior of the critical gel, which expresses itself in the self-similar relaxation, several different relationships between the critical exponent n and the fractal dimension df have been proposed recently. The fractal dimension ds of the polymer cluster is commonly defined by [16,42]... 

Muthukumar and Winter [42] investigated the behavior of monodisperse polymeric fractals following Rouse chain dynamics, i.e. Gaussian chains (excluded volume fully screened) with fully screened hydrodynamic interactions. They predicted that n and d (the fractal dimension of the polymer if the excluded volume effect is fully screened) are related by... 

The power-law variation of the dynamic moduli at the gel point has led to theories suggesting that the cross-linking clusters at the gel point are self-similar or fractal in nature (22). Percolation models have predicted that at the percolation threshold, where a cluster expands through the whole sample (i.e. gel point), this infinite cluster is self-similar (22). The cluster is characterized by a fractal dimension, df, which relates the molecular weight of the polymer to its spatial size R, such that... 

The fractal dimension measures how open or packed a structure is lower fractal dimensions indicate a more open system, while higher fractal dimensions indicate a more packed system (22). Theories relating the fractal dimension to the relaxation exponent, n, have been put forward and these are based on whether the excluded volume of the polymer chains is screened or unscreened under conditions near the gd point (23). It is known that the excluded volume of a polymer chain is progressively screened as its concentration is increased, the size of the chain eventually approaching its unperturbed dimensions. Such screening is expected to occur near the... 

Keywords. Solution properties. Regularly branched structures. Randomly and hyperbranched polymers. Shrinking factors. Fractal dimensions. Osmotic modulus of semi-di-lute solutions. Molar mass distributions, SEC/MALLS/VISC chromatography... 

P(r) can be transformed into a distribution of the particle size as defined by the hydrodynamic radius Rh. But only for TDFRS, and not for PCS, a particle size distribution in terms of weight fractions can be obtained without any prior knowledge of the fractal dimension of the polymer molecule or colloid, which is expressed by the scaling relation of Eq. (39). This can be seen from the following simple arguments ... 

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. 

For Q<0, this distribution function is peaked around a maximum cluster size (2Q/(2Q-1))< >, where < > is the mean cluster size. 2Q=a+df1 is a parameter describing details of the aggregation mechanism, where a1 is an exponent considering the dependency of the diffusion constant A of the clusters on its particle number, i.e., A NAa. This exponent is in general not very well known. In a simple approach, the particles in the cluster can assumed to diffusion independent from each other, as, e.g., in the Rouse model of linear polymer chains. Then, the diffusion constant varies inversely with the number of particles in the cluster (A Na-1), implying 2Q=-0.44 for CCA-clusters with characteristic fractal dimension d =l.8. 

Daoud M, Stanley HE and Stauffer D, "Scaling, Exponents and Fractal Dimensions", In Mark JE (Ed), "Physical Properties of Polymers Handbook", 2nd Ed, Springer-Verlag, Berlin, 2007, Chap. 6. 

The interaction between the water and the polymer occurs in the vicinity of the polymer chains, and only the water molecules situated in this interface are affected by the interaction. The space fractal dimension da is now the dimension of the macromolecule chain. If a polymer chain is stretched as a line, then its dimension is 1. In any other conformation, the wrinkled polymer chain has a larger space fractal dimension, which falls into the interval 1 < d( < 2. Thus, it is possible to argue that the value of the fractal dimension is a measure of polymer chain meandering. Straighter (probably more rigid) polymer chains have da values close to 1. More wrinkled polymer (probably more flexible) chains have da values close to 2 (see Table III). 

Mass fractal dimensions are always less than the dimension of the space in which the fractal object exists. Therefore, as a fractal structure grows, its mass increases less rapidly than the volume it occupies. Therefore, the density of a fractal object is not constant but decreases with increasing size. Surface fractal dimensions, on the other hand, must lie in the range of one less than the dimension of space up to the dimension of space. The surface area of these objects increases with increasing mass at a faster rate than for Euclidian objects. As a result, the surfaces are very convoluted. Both types of structures are observed in silica polymers. 

Much of the current interest in fractal geometry stems from the fact that fractal dimensions are experimentally accessible quantities. For polymers and colloids, the measurement techniques of choice are scattering experiments using X-rays, neutrons, or light. These measurements may be made on liquids or solids and can be performed readily as a function of time and temperature. Both mass and surface fractal structures yield scattering curves that are power laws, the exponents of which depend on the fractal dimension (6). For mass fractal structures the relation is... 

The concept of a fractal dimension enables the structures of silicate species to be divided into two classes (1) those with a dimension less than 3, which may be considered to be true polymers, and (2) those with a dimension of 3, which are perhaps best described as colloidal particles (5). The colloidal class may be further subdivided into particles with fractally rough surfaces (D > 2) and particles with smooth surfaces (D = 2). In this chapter, the term polymer will be used for any product of the condensation reaction, regardless of fractal dimension, although the term polymeric will be used for structures with dimensions less than 3, and the term colloidal will be used for structures with dimensions equal to 3, in keeping with the terminology used in the literature (9). 

To assign physical meaning to D, we associate D with the mass fractal dimension of the protein, defined by M LD. Then D is the slope of log M versus log L, where M is the mass of all protein atoms enclosed by concentric spheres of radius L. For a completely collapsed, space-filling polymer, D is 3, but D may be generally less than 3 in proteins due to a combination of compact structures and spacing between them. Figure 14 shows log M versus log L for cytochrome c. The concentric spheres of radius L are centered on one of the Ca s of the protein backbone. We compute M for values of L from 2 A to 16 A. The plot appears linear with slope 2.15. Continuing in this way, we compute... 

In Chapters 2, 3, 5, and 6, the fractal dimension D of polymers will be derived in different conditions. Examples of the fractal dimensions of polymers are shown in Table 1.2. 

---