Models square-lattice network

Figure 2.9.3 shows typical maps [31] recorded with proton spin density diffusometry in a model object fabricated based on a computer generated percolation cluster (for descriptions of the so-called percolation theory see Refs. [6, 32, 33]).The pore space model is a two-dimensional site percolation cluster sites on a square lattice were occupied with a probability p (also called porosity ). Neighboring occupied sites are thought to be connected by a pore. With increasing p, clusters of neighboring occupied sites, that is pore networks, begin to form. At a critical probability pc, the so-called percolation threshold, an infinite cluster appears. On a finite system, the infinite cluster connects opposite sides of the lattice, so that transport across the pore network becomes possible. For two-dimensional site percolation clusters on a square lattice, pc was numerically found to be 0.592746 [6]. 

Figure 1.4 A 6 X 6 square lattice site model. The dots correspond to multifunctional monomers. (A) The encircled neighboring occupied sites are clusters (branched intermediate polymers). (B) The entire network of the polymer is shown as a cluster that percolates through the lattice from left to right.

A second tutorial example may be useful. Suppose that an extended communication network, modeled as a large two-dimensional square-lattice grid connected to heavy bars at two opposite boundaries, is attacked by a stochastic saboteur, who, with wire-cutters, severs the grid interconnections. What fraction of the links must be cut in order to isolate the two bars from each other The answer, given by a a percolation... 

In the case of a concentrated polymer network modeled by the square lattice (inset in Figure 12.26) a solution for the order parameter profile (see Figure 12.28) is found numerically [74]. It can be seen that for thin fibrils... 

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