Percolation fractal

Figure 4.11 A percolation fractal embedded on a 2-dimensional square lattice of size 50 x 50. Cyclic boundary conditions were used. We observe, especially on the boundaries, that there are some small isolated clusters, but these are not isolated since they are actually part of the largest cluster because of the cyclic boundary conditions. Exits (release sites) are marked in dark gray, while all lighter grey areas are blocked areas. Reprinted from [87] with permission from American Institute of Physics.

The drug molecules move inside the fractal matrix by the mechanism of diffusion, assuming excluded volume interactions between the particles. The matrix can leak at the intersection of the percolation fractal with the boundaries of the square box where it is embedded, Figure 4.11. 

Figure 4.13 Plot of the number of particles (normalized) remaining in the percolation fractal as a function of time t for lattice sizes 100 x 100, 150 x 150, and 200 X 200. n (t) is the number of particles that remain in the lattice at time t and no is the initial number of particles. Simulation results are represented by points. The solid lines represent the results of nonlinear fitting with a Weibull function.

The power laws for viscoelastic spectra near the gel point presumably arise from the fractal scaling properties of gel clusters. Adolf and Martin (1990) have attempted to derive a value for the scaling exponent n from the universal scaling properties of percolation fractal aggregates near the gel point. Using Rouse theory for the dependence of the relaxation time on cluster molecular weight, they obtain n = D/ 2- - Df ) = 2/3, where Df = 2.5 is the fractal dimensionality of the clusters (see Table 5-1), and D = 3 is the dimensionality of... 

The heterogeneous nature of polymer melts at Tgtwinkling fractal theory (TFT) [Wool, 2008a,b]. Wool considers Tg to result from the molecular cooperativity that leads to dynamic percolating fractal structures below Tc. He assumes Boltzmann distribution of diatomic oscillators interacting via the Morse anharmonic potential. Integrating the latter from zero to the inflection point, he expresses the T dependence of solidified polymer fraction as... 

Nolte DD, et al. The fractal geometry of flow paths in natural fractures in rock and the approach to percolation. Fractals in Geophysics. 1989. 

Meakin, R Stanley, H. E. Novel dimension-independent behavior for diffusive annihilation on percolating fractals. J. Phys. A, 1984,17( 1), L173-L177. 

Here we have very briefly introduced the (lattice) SAW model of linear polymers, their configurational statistics and the (lattice) percolation model of disodered media. Approximate mean field-like and scaling arguments have been forwarded to indicate that the SAW critical behaviour on disordered lattices, percolating lattice in particular, could be significantly different from those of SAWs on pure lattices. More careful analysis, as we will see in the following chapters, show even more subtle effects of disoder on the polymer conformation statistics. Also, as we will see, such effects are not necessarily confined only to the cases of extreme disorder like percolating fractals. 

Hierarchy can be described in analogy to rope (stretched polymer molecules in domains that make up nanofibers, combined to microwhiskers, bundled into fibers that are spun into yarn that is twined to make up the rope). Wood and tendon are biological examples that have six or more hierarchical levels. Compared to these, fiber-reinforced matrix composites made up of simple massive fibers embedded in a metallic, ceramic, or polymer matrix are primitive. Hierarchical inorganic materials, as discussed in Chapter 7, can be made with processes for fractal-like solid products spinodal decomposition, diffusion-limited growth, particle precipitation from the vapor, and percolation. Fractal-like solids have holes and clusters of all sizes and are therefore hierarchical if the interactions... 

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