Fractal dimension table

The monolayer amount adsorbed on an aluminum oxide sample was determined using a small molecule adsorbate and then molecular-weight polystyrenes (much as shown in Ref. 169). The results are shown in the table. Calculate the fractal dimension of the oxide. 

Obviously the roughness factor is similarly arbitrary, but it is of interest to use Eq. 25 to compute its value for some trial values of D and a. This is done in Table 2. In order to map the surface features even crudely, the probe needs to be small. It can be seen that high apparent roughness factors are readily obtained once the fractal dimension exceeds 2, its value for an ideal plane. 

Table 17.1 also displays the mass-fractal dimension, df, given by the power-law value, P, for the second level (aggregate). The df= 1.63 for A1 is less than that, and is less than 2.37 for A2 and 2.1 for A3. The lower value of df could indicate a more open structure of the A1 sample, " but it could also arise from less interpenetration of the silica aggregates occurring during drying. ... 

Physically, the wetting abdity increases (the contact angle decreases) as the values of the fractal dimension of the electrode increases if the electrode material is same. However, in this study, we could not obtain a good correlation between the fractal dimensions and the wetting abilities as shown in Table 1. It means that not only the physical properties such as the surface irregularity and roughness but also the chemical interaction between electrolyte and electrode were important in wetting ability. 

Table 1 Surface fractal dimension determined by nitrogen adsorption...

The self-similar fractal dimension determined from Figure 9 by Eq. (35) is listed in Table 1 (3rd column). The values of d7 ss... 

V . The resulting fractal dimensions are listed with the dimension of the flat film surface in Table 2 (3rd column). 

Table 6. Relationship between the fractal dimension dp the exponent for the molar mass dependence of the second virial coefficient and the expected exponent m for the osmotic modulus when the scaling assumptions of Eqs. (93)-(96) are made. The experimental data were derived from the exponents for the second virial coefficient...

Figure 2.13. Determination of the fractal dimension (D — 2.40 and D = 2.43) for 3rd (top) and 4th (bottom) generation amide-based dendrimers. [See the table presented in Figure 2.8]...

Light-scattering experiments on flocculating suspensions of silica colloids provided the data in the following table. (R is the average cluster radius.) Estimate the fractal dimension of the clusters formed and indicate whether the flocculation process is transport or reaction controlled. Hint Apply Eq. 6.1 and the concepts in Section 6.1.)... 

In the following table are data on the dependence of the average floccule radius, achieved after 500 s of flocculation, on the initial primary particle number density for a transport-controlled flocculation process. Estimate the fractal dimension of the floccules formed. (Answer D 1.8, based on a log-log plot.)... 

TABLE 2 Fractal Dimension Estimates in Colloidal Systems... 

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). 

Table 6.2. Values of accessible fractal dimension evaluated from adsorption data (taken from Farin and Avnir, 1989)...

Table 17.7 summarizes the effects of the microstructural factors on the microscopy fractal dimensions, Dj, y, and Zlpr- Different fractal dimensions reflect different aspects of the microstructure of the fat crystal networks and thus have different meanings. It is necessary to define which structural characteristic is most closely related to the macroscopic physical property of interest (mechanical strength, permeability, diffusion) and then use the fractal dimension that is most closely related to the particular structural characteristic in the modeling of that physical property. 

Table 17.7. The effect of the microstructural factors on the microscopy fractal dimensions, D >, Df, and Dfj.

TABLE 3 Fractal Dimension, Df, from Different Aggregation Models"... 

TABLE 6.4 Effective Fractal Dimension Obtained from Reorganization of 3-D Cluster-Cluster Aggregates"... 

The fractal dimension of an aggregate depends on the mode of aggregation, as shown in Figure 10.35 [77,78] and Table 10.10. For a rapid. 

TABLE 10.10 Fractal Dimension of Huee-Dimensional Aggr ate (Le., d = 3) Growth Model... 

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