Square lattice/networks

However, two-dimensional networks appear to capture almost all of the important physics and chemistry of the problem. (Their dimensionality, two dimensions instead of the three of a real porous medium, is fundamentally incorrect.) Figure 6 illustrates a square-lattice network in which all tubes have the same length and connectivity but different radii. Important parameters for a network include the population distribution of radii, the physical distribution of those radii in the medium, and the connectivity (number of tubes that meet at a node). 

Figure 6. Two-dimensional, square-lattice network with connectivity four and a distribution of capillary radii.

Fig. 4 Elastic moduli of two types of cable network, (a) A triangular lattice network and designated test region (green) in the middle, (b) Network in (a) is stretched horizontally, (c), (d) Unstretched and stretched square lattice network, (e) Young s modulus E determined from the stretching experiment for both types of network, plotted as a function of number of nodes in the test region. Note that, a prestressed network has a larger Young s modulus, (d) Poisson s ratio a for triangular and square lattice networks. Clearly, prestress reduces the Poisson s ration for both types of 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]. 

Each node is drawn at a position defined by its two weights, interpreted as an x- and a y-coordinate, respectively. Connecting lines are then drawn to join nodes that are next to each other in the SOM lattice. Thus, if the first and second SOM nodes, with lattice positions [0,0] and [0,1], have initial weights (0.71,0.06) and (0.98,0.88), points are drawn at (x = 0.71, y = 0.06) and (x = 0.98, y = 0.88) and connected with a line. The points occupy the available space defined by the range of x and y coordinates. Because the data points are positioned at random within a 1 x 1 square, the network nodes are initially spread randomly across that same space. 

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... 

Figure 6.9 pictures the example of the triangular-square network taken from [6.38]. Some 4-coordinated sites are seen comprising inner boundaries of this LRC. It is easy to notice that there are two 5-coordinated atoms in the first coordination sphere of each 5-coordinated atom in LRC. This circumstance follows from the fact that, in P-polyhedra, we have an even number of squares leading to the formation of the MRO, which manifests itself in the formation of chains of 5-coordinated atoms. Collins and Kawamura studied the thermodynamic properties of triangular-square lattices. Kawamura established the existence of the first-order phase transition connected with the transformation of the crystalline structure into a topologically disordered one. 

Charlaix et al. (1988) also conducted a study of NaCl and dye transport in etched transparent lattices. A fully connected square lattice with a lognormal distribution of channel widths and a partially connected hexagonal lattice (a percolation network) were considered. They concluded that the disorder and heterogeneity of the medium determined the characteristic dispersion length. From experimental data on the percolation network, they showed that this dispersion length was close to the percolation correlation length, p. 

We use now the 3-dimensional cubic network to simulate deactivation. In a three dimensional structure there is higher possibility of forming internal clusters, connected by fewer channels with the rest of the structure. The blocking of channels and or cavities affects these regions more drastically. The connectivity can also be more readily varied, 0 to 6 compared 0 to 4 in a two dimensional square lattice. The bias in size distributions of cavities and channels can be studied, besides the effect of correlation, Q. 

Simulations on the effect of step free energy on grain growth behaviour have also been made. Figure 15.11 shows the result of a Monte Carlo simulation made by Cho. For the simulation, Cho assumed that the grain network was a set of grains with a Gaussian size distribution (standard deviation of 0.1) located on vertices of a two-dimensional square lattice. Deterministic rate equations, Eq. (15.15) for v/> and Eq. (15.29) for v j, were... 

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... 

A percolative picture emerges from these results [8, 30, 32, 35, 55, 56]. As the density of polarons is increased, they start to become in contact. If the polarons are single-site polarons, charges will remain localized even if two polarons are in contact because of the high coulomb repulsion to place two carriers on the same site (Hubbard U). However, if the polarons are more extended over several sites, as is the case for (La/Sr)Mn03 shown above, charge carriers will become mobile within the connected network of the polarons. If the size of the network of the connected polarons reaches a macroscopic scale, metallic conduction commences. At the M-I transition the number of short Mn—O bonds is about 5, as shown in Fig. 10, indicating that the volume fraction of the undistorted, metaUic sites is 50%. This is consistent with the percolation in a two-dimensional square lattice [57]. 

Zeolite lattices have a network of very small pores. The pore di uneter of nearly all of today s FCC zeolite is approximately 8.0 angstroms (°A). These small openings, with an internal surface area of roughly 600 square... 

Fig. 23. Plot of the dipolar broadening parameter AG (left scale) and its relation to the square root of rigid lattice second moment, AM2 = 15 kHz (right scale), for polystyrene networks crosslinked with DVB (solid symbols) and EDM (open symbols) swollen to equilibrium, vs l/n, the reciprocal nominal number of C—C bonds between crosslinks points. Solvents CC14 ( ), CDC13 (A),...

The origins of percolation theory are usually attributed to Flory and Stock-mayer [5-8], who published the first studies of polymerization of multifunctional units (monomers). The polymerization process of the multifunctional monomers leads to a continuous formation of bonds between the monomers, and the final ensemble of the branched polymer is a network of chemical bonds. The polymerization reaction is usually considered in terms of a lattice, where each site (square) represents a monomer and the branched intermediate polymers represent clusters (neighboring occupied sites), Figure 1.4 A. When the entire network of the polymer, i.e., the cluster, spans two opposite sides of the lattice, it is called a percolating cluster, Figure 1.4 B. 

---