Square lattice percolation

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]. 

We study here the A + 5B2 —> 0 reaction upon a disordered square lattice on which only a certain fraction S of lattice sites can be accessed by the particles (the so-called active sites). We study the system behaviour as a function of the mole fractions of A and B in the gas phase and as a function of a new parameter S. We obtain reactive states for S > Sq where Sq is the kinetically defined percolation threshold which means existence of an infinite cluster of active sites. For S < Sq we obtain only finite clusters of active sites exist. On such a lattice all active sites are covered by A and B and no reaction takes place as t —> 00. 

If S is smaller than the site percolation threshold for the square lattice Sc = 0.59275 we obtain a system which consists only of finite clusters. In principle these clusters can be completely occupied by one-kind species. For the case that no desorption is allowed this represents a poisoned state for which the production rate Rco2 goes to zero as t —> oo. Then the whole system consists of finite clusters poisoned by particles A or B. For this state the condition Ca + Cb = S holds where C is the density of particles of type A (C a + Cb + Co = S). 

We have studied above a model for the surface reaction A + 5B2 -> 0 on a disordered surface. For the case when the density of active sites S is smaller than the kinetically defined percolation threshold So, a system has no reactive state, the production rate is zero and all sites are covered by A or B particles. This is quite understandable because the active sites form finite clusters which can be completely covered by one-kind species. Due to the natural boundaries of the clusters of active sites and the irreversible character of the studied system (no desorption) the system cannot escape from this case. If one allows desorption of the A particles a reactive state arises, it exists also for the case S > Sq. Here an infinite cluster of active sites exists from which a reactive state of the system can be obtained. If S approaches So from above we observe a smooth change of the values of the phase-transition points which approach each other. At S = So the phase transition points coincide (y 1 = t/2) and no reactive state occurs. This condition defines kinetically the percolation threshold for the present reaction (which is found to be 0.63). The difference with the percolation threshold of Sc = 0.59275 is attributed to the reduced adsorption probability of the B2 particles on percolation clusters compared to the square lattice arising from the two site requirement for adsorption, to balance this effect more compact clusters are needed which means So exceeds Sc. The correlation functions reveal the strong correlations in the reactive state as well as segregation effects. 

K. M. Middlemiss, S. G. Whittington, and D. S. Gaunt,/. Phys. A, 13, 1835 (1980). Monte Carlo Study of the Percolating Cluster for the Square Lattice Site Problem. 

Figure 4.15. Random square lattice of sites (A) and bonds (B), corresponding to the same concentration of passing bonds (solid lines). The concentration of sites is about 71% in lattice A, and that of bonds 50% in both lattices. Thus, the B lattice is at a percolation concentration, whereas the A lattice is above the percolation threshold.176-177 Note the aggregation of bonds in lattice A relative to lattice B.

The numerical solution of the system (4.86), by a procedure of the Newton-Raphson type with two variables, requires the calculation of the derivatives dtjds and dt /5Wc . The results we obtained for a square lattice are similar to those by Yonezawa and Odagaki180 for a cubic lattice. The most striking feature is the existence, at low concentration, of a gap in the density of states,179 which isolates the zero energy on which a 3 peak builds up. Thus the HCPA produces a forbidden region of energy for the transport the gap and the 3 peak disappear at a critical concentration, analogous to the percolation threshold of the mean-field of resistances. 

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.

In the model of bond percolation on the square lattice, the elements are the bonds formed between the monomers and not the sites, i.e., the elements of the clusters are the connected bonds. The extent of a polymerization reaction corresponds to the fraction of reacted bonds. Mathematically, this is expressed by the probability p for the presence of bonds. These concepts can allow someone to create randomly connected bonds (clusters) assigning different values for the probability p. Accordingly, the size of the clusters of connected bonds increases as the probability p increases. It has been found that above a critical value of pc = 0.5 the various bond configurations that can be formed randomly share a common characteristic a cluster percolates through the lattice. A more realistic case of a percolating cluster can be obtained if the site model of a square lattice is used with probability p = 0.6, Figure 1.5. Notice that the critical value of pc is 0.593 for the 2-dimensional site model. Also, the percolation thresholds vary according to the type of model (site or bond) as well as with the dimensionality of the lattice (2 or 3). 

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.

Note that the classic SLSP model for a square lattice ABCO would provide a percolation trajectory connecting opposite sites OA and CB (see Fig. 32). On the other hand, for visualization of dynamic percolation we shall consider an effective three-dimensional static representation of a percolation trajectory connecting ribs OA and ED spaced distance Lh (see Fig. 33). Such a consideration allows us to return to the lattice OEDA, with the initial dimension <7 = 2, which is non-square and characterized by the two dimensionless sizes Lh 11 and L/l. The new lattice size of the system can be determined from the rectangular triangle OEQ by... 

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... 

P is the key function characterizing a percolation process, and here it plays the role of the order parameter used to describe order-disorder phenomena and phase transitions. Its behavior for a square lattice is show in Fig. 39. 

Fig. 41. Percolation probability for finite-sized lattices. Computer calculations of the percolation probability, P(p), as a function of the site-filling probability, p, for two-dimensional square lattices of varied dimension O, 10 x 10 , 20 x 20 , 40 x 40. Each curve is an average over a set of site percolation simulations for a lattice size. The site percolation threshold for an infinite two-dimensional square lattice is 0.593. Nonzero values of P p) below the infinite lattice threshold reflect the variance of the threshold value for finite lattices (unpublished results).

Figure 41 shows the percolation probability P(p), determined by averaging Monte Carlo simulations for site percolation on a two-dimensional square lattice, for finite lattices of varied size. For an infinite lattice of this type, pc = 0.593. The nonzero values of P p) below p = 0.593 reflect the dispersion in pc found for finite lattices. A protein, with several hundred water sites on its surface, would fall in the range of lattice sizes modeled in Fig. 41. The shape of the P(p) function is not strongly affected by lattice size.

A quantitative analysis of the failure process was made by Duxbury et al (1987) by modelling the system by a lattice and we shall present their results. The simplest lattices were taken a square lattice in two dimensions and a simple cubic lattice in three dimensions, in which the bonds are all equal resistors to begin with. Each resistor can stand a current up to io. If i > io, the resistor is fused and becomes a perfect insulator. It is believed that the results are not dependent on the type of the lattice as it was proven in the case of percolation. The size of the lattice is L in two dimensions it... 

The properties of the mean shortest path has been studied theoretically near the percolation threshold (Chayes et al 1986, Stinchcombe et al. 1986) and numerically in the whole range of p (Duxbury et aL, unpublished). The important property is the variation of = (no)/T, the mean number of resistances in the shortest path, with p. For p near 1, g decreases from 1 linearly with a slope dg/dp about 3 (for the square lattice). In the vicinity oiPc, g goes to zero as (p —Pc) where i/ is the correlation length exponent. 

Manna and Chakrabarti (1987) performed the same kind of calculation for site percolation in a square lattice for the entire range of p (0 < p < Pc) and at the same time determined the minimum gap g (or the shortest path). These results are shown in Fig. 2.15. Near Pc, they found that both E and g go to zero with almost the same exponent value equal to about unity. This is still consistent with the above theoretical analysis. We point out that Bowman and Stroud worked with L = 150, while the results of Manna and Chakrabarti are for L = 25 only (both in d = 2). The smaller size of the sample limits the possibility to reach values of p near enough to Pc and... 

Fig. 4. Realization of bond percolation on a 50 X 60 section of the square lattice for four different values of q. The diagrams have been created using the same sequence of pseudorandom numbers, with the result that each graph is a subgraph of the next. Attentive readers may verify that open paths exist joining the left to the right side when = 0.51 but not at q = 0.49. (From Ref. 13, with permission.) continued)...

Table X present results obtained for the overage coordination number (Z) and for the average number of complete bonds around a lattice site at the percolation threshold (i.e., the product of (Z) and p at the critical point) for the lattices considered. Thus, for the inhomogeneous lattices considered this product can be much different from the value 2 which has been obtained for the infinite uniform square lattice [3].

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. 

---