Fractal number density

The porous structure of a cluster fractal can be quantified by estimating its number density at any stage of growth. For example, the number density of any cluster in Fig. 6.10 can be calculated with the equation... 

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.)... 

Measurements of the average floccule radius in a suspension of colloids, using light-scattering techniques, indicated the following time dependence R(t) = Ro(l + yty, where Ro = 5 nm, y = 9.3 s 1, and j8 = 0.56. Derive this equation, and estimate the initial number density of primary particles as well as the fractal dimension of the floccules formed. (Answer p0 = 7.5 x 1017 m 3 and D = 1.78.)... 

At low relative pressures p/p0 or thin adsorbate films, adsorption is expected to be dominated by the van der Waals attraction of the adsorbed molecules by the solid that falls off with the third power of the distance to the surface (FHH-regime, Eq. 3a). At higher relative pressures p/p0 or thick adsorbate films, the adsorbed amount N is expected to be determined by the surface tension y of the adsorbate vapor interface (CC-regime, Eq. 3b), because the corresponding surface potential falls off less rapidly with the first power of the distance to the surface, only. The cross-over length zcrit. between both regimes depends on the number density np of probe molecules in the liquid, the surface tension y, the van der Waals interaction parameter a as well as on the surface fractal dimension ds [100, 101] ... 

The chains in a monodisperse two-dimensional melt are roughly the size of a thermal blob and are therefore barely ideal. The number of the other chains in a pervaded area of a given chain is the product of this area Nb and the two-dimensional number density of chains 1 jNb and is of the order of unity (Pss 1). Thus, chains do not significantly interpenetrate each other in two dimensions. This is the expected result whenever the fractal dimension of the object and the dimension of space are the same. 

We now provide an example of such an inversion from the work of Wright et al (1992) in which spatial computer simulations were used to generate data on the aggregation of fractal clusters formed by Brownian motion of colloidal particles. We consider three-dimensional diffusion under two circumstances (i) that in which the diffusion coefficient of the cluster is independent of its mass and (ii) that in which the diffusion coefficient, decreases with increasing mass. The simulated process automatically produces noisy data and the number density in cluster mass is presented in Figure 6.2.10 at three different times for both cases (i) and (ii). 

FIGURE 6.2.10 Number density of fractal clusters as a function of cluster mass at different times obtained from computer simulations (i) mass-independent diffusion (ii) mass-dependent diffusion. (From Wright et al, 1992.)... 

The generation parameter defining the generation of ionizing trajectories in the self-similar structure in Fig. 10 is related to the number w of encounters of the two electrons at ri = T2 rather than to the ionization time. This interpretation is confirmed in Fig. 11 which shows the density n of trajectories starting with initial conditions uniformly distributed in the middle panel of Fig. 10 as function of the number w of encounters of the two electrons and of the ionization time T. The density n is proportional to minus the derivative of the survival probability with respect to the relevant variable (w or T). The logarithmic plot in Fig. 11a reveals an exponential decay of the density, n(w)ocexp(—0.27w), and hence also of the survival probability, as a function of the number of encounters of the two electrons, just as expected for a self-similar fractal set of trapped trajectories. The doubly logarithmic plot of the density of trajectories in Fig. 11b reveals a power-law decay of the density, (T) oc and hence... 

Fractal dimension, D is considered as an effective number that characterises the irregular electrode surface. The term has been related to physical quantities such as mass distribution, density of vibrational stages, conductivity and elasticity. If we consider a 2-D fractal picture in its self-similar multi-steps, one can draw various spheres of known radii at various points of its structure and may thus count the number of particles, N inside the sphere by microscope, following relation will then hold good ... 

To get an expression for d, we need to explore the relationship between the cross-section of the islands in a given layer with film coverage. If 5 represents the area of a two-dimensional cluster and D its fractal (Hausdorff) dimension, then the island cross-section is proportional to sp, where Pi = HD. D = 2 corresponds to a compact island, but for pentacene that grows by diffusion limited aggregation (DLA), D 513. Noting from before that, prior to coalescence the island density, n remains constant and composed of identical islands, then the total cross-section of a layer becomes d = n SP. Since 9 = n S, then Qp. Similarly, if after coalescence the number of holes in the film also remains constant, then the perimeter of each layer can be related to its coverage as shown in Figure 5.1.12b and c by [29] ... 

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