Fractal radius

As a result of the rapid decrease in density as a function of increasing fractal radius, the fractal density may become so low as to be non-self-supporting. As a consequence, there is be a limiting, or critical, fractal radius, r beyond which particle growth is not favored or greatly retarded. 

The relative density is a measure of the volume element bounded by ro and r Although r represents the critical fractal radius, the relative density is not zero at that point, inasmuch as it approached zero asymptotically. Therefore,... 

Figure 3 Calculated critical fractal radius versus fractal dimension for a 0.1 critical fractal relative density.

A primary sol particle in an acid-cataly2ed sol has radius between 1 and 2 nm (3). The secondary fractal particle has a radius, R, of 5 to 20 nm as seen from saxs (3). For the TMOS-based sols investigated by saxs, ( increases with time, as does the Guinier radius, R. The stmcture reaches a fractal dimension around 2.3 at the gelation point. 

One characterizes a fractal cluster by its fractal dimension Df which can be determined from the integration of the radius-dependent mass density p r) from the center up to a large radius R as follows ... 

Rupture of fractal (flocculated) aggregates of polystyrene latices in simple shear flow and converging flow was studied by Sonntag and Russel (1986, 1987b). For simple shear flow and low electrolyte concentrations, the critical fragmentation number decreases sharply with agglomerate radius (R) as... 

The dissolution channels (wormholes), obtained under certain conditions of attack of carbonate rocks by hydrochloric acid, have been recently proven to have a fractal geometry. An equation was proposed, relating the increase of the equivalent wellbore radius (i.e. the decrease of the skin) to the amount of acid injected, in wellbore geometry and in undamaged primary porosity rocks. This equation is herein extended to damaged double porosity formations through minor modifications. 

Fig. 33 Scaling of the molar mass of PNIPAM mesoglobules (M 8) vs. their radius of gyration (i g) with fractal dimensionality 2.7 (filled symbols) and the shape factor J g/J h (open symbols). The conditions at which mesoglobules were formed correspond to those in Table 2 Mw = 27300gmol 1, non-equilibrium heated (circles) Mw = 160000gmol 1, nonequilibrium heated (triangles) Mw = 160000gmoL1, equilibrium heated (squares). (Reprinted with permission from Ref. [ 147] copyright 2005 Elsevier)...

There are two molecular probe methods available for the determination of surface fractal dimension. One is the multiprobe method (MP method),83,84,87-100 which uses several kinds of multiprobe molecules with different molecular sizes and requires the number of adsorbed molecules to form a monolayer Nmoao for each probe molecule. If the probe molecule is varied through a series of spheres with radius rm, the surface fractal dimension is given by Eq. (7) ... 

It is known [3], that macromolecular coil in various polymer s states (solution, melt, solid phase) represents fractal object characterized by fractal (Hausdorff) dimension Df. Specific feature of fractal objects is distribution of their mass in the space the density p of such object changes at its radius R variation as follows [4] ... 

Since the mean square radius of gyration requires a z-average but the molar mass a weight average the fractal dimension remains unchanged only if the ratio is independent of the molar mass or close to unity. These conditions are mostly fulfilled with polydisperse linear chains but not for the randomly branched ones. Here this ratio increases strongly with the molar mass. 

Thus the molar mass dependence of the CP-parameter can be estimated when, besides the exponent, the fractal dimension of the clusters could be measured. This fractal dimension can be obtained from the molar mass dependence of the radius of gyration of the fractions, or from the angular dependence of the... 

Note 1 w oc r in which m is the mass contained within a radius, r, measured from any site or bond within a fractal structure. 

The development leading to Equation (66) in this section related the form factor P(6) to the radius of gyration Rg, which is one measure of the structure of a particle. We can actually get much more information from the form factor. In the following section, we discuss this and illustrate the use of P(d) for measuring the fractal dimension (defined in Chapter 1, Section 1.5b. 1) of an aggregate. 

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 ... 

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