Lattice models random bond model

The transition to a model of random bonds neglects the correlations imposed by construction between passing bonds around an A site, as is shown distinctly in Fig. 4.15, where two lattices satisfying the equivalence (4.83) are compared one lattice with random bond distribution, and one lattice with random site suppressions. Aggregation of bonds is easily discernable in the case of site percolation. However, as a matter of fact, these correlations have no importance in the case of conductivity, so that we may obtain a good approximation when leaving them out.178 It will be shown below that this approximation is questionable for more local properties where the microscopic arrangements of bonds may be crucial. 

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). 

As mentioned before, the disordered solids will be mostly modelled in this book using randomly diluted site or bond lattice models. A knowledge of percolation cluster statistics will therefore be necessary and widely employed. Although this lattice percolation kind of disorder will not be the only kind of disorder used to model such solids, as can be seen later in this book, the widely established results for percolation statistics have been employed successsfully to understand and formulate analytically various breakdown properties of disordered solids. We therefore give here a very brief introduction to the percolation theory. For details, see the book by Stauffer and Aharony (1992). 

In the lattice model of random dielectrics, we have considered, so far, dielectric bonds with fixed breakdown threshold. We now consider the problem of breakdown of a random dielectrics, where the bonds have random break-... 

The important open question is precisely what happens just after the first bond is fused or broken. There is no precise answer, and it seems that the efforts in the future will be concentrated in this direction. We reported that the current belief is that in the lattice models, after the failure of the first bond, a cascading effect occurs and the failure propagates, even when the current or the voltage across the sample is kept constant at the first failure value. However, this dynamic problem has not been solved yet. Very recently of course Zapperi et al. (1997) have confirmed the critical divergence of breadown susceptibility x the breakdown point in the random fuse some other models, as discussed in the section 2.3.7(b). They also argued that this divergence of % = / nP n)dn " suggests... 

In the discrete lattice model, discussed above, each bond is identical, having identical threshold values for its failure. In the laboratory simulation experiments (discussed in the previous section) on metal foils to model such systems, holes of fixed size are punched on lattice sites and the bonds between these hole sites are cut randomly. If, however, the holes are punched at arbitrary points (unlike at the lattice sites as discussed before), one gets a Swiss-cheese model of continuum percolation. For linear responses like the elastic modulus Y or the conductivity E of such continuum disordered systems, there are considerable differences (Halperin et al 1985) and the corresponding exponent values for continuum percolation are higher compared to those of discrete lattice systems (see Section 1.2.1 (g)). We discuss here the corresponding difference (Chakrabarti et al 1988) for the fracture exponent Tf. It is seen that the fracture exponent Tf for continuum percolation is considerably higher than that Tf for lattice percolation Tf = Tf 4- (1 -h x)/2, where x = 3/2 and 5/2 in d = 2 and 3 respectively. 

Stanley and Teixeira have introduced a new polychromatic correlated-site percolation model [4], which has the novel feature that the partitioning of the sites into different species arises from a purely random process - that of random bond occupancy. By polychromatic one thus means that each lattice site is differently colored according to bond occupancy. 

We consider in this book the problem of polymer chain statistics in a disordered (say, porous) medium. If the porous medium is modelled by a percolating lattice [5], we can consider the following problem let the bonds (sites) of a lattice be randomly occupied with concentration p (> Pc, the percolation threshold) the SAWs are then allowed to have their steps only on the occupied bonds (through the occupied sites). We address the following questions [6,7] does the lattice irregularity (of the dilute lattice) affect the SAW statistics ... 

Randomness is introduced by allowing the interaction energy to be random on each and every bond. The first model, model A, has independent random energy on all the 2b bonds. The randomness in the second model, model B, is taken only along the longitudinal direction so that equivalent bonds on all directed paths have identical random energy. Model B is a hierarchical lattice version of the continuum RANI model. 

Thus, in the case of solid solutions of sp Iqrbrid binary compoimds, Vegard s law applies to the average values of the bond lengths d, whereas d and d2 remain independent of the composition of the solid solution. The ideal lattice model with a random distribution of atoms over the sites is strictly applicable to solid solutions only if the values of d and d2 are similar, which implies that it is limited to solid solutions consisting of members of Goldschmidt s iso-electronic series, or members of groups of compounds classified in [9,10]. In all other cases, this approach is only very approximate. 

Although there are probably other universality classes, this transition was successfully modeled by bond percolation [6]. Generally, bond percolation on a lattice has each bond (line connecting two neighboring lattice sites) present randomly with probability p and absent with probability 1-p. Clusters are groups of sites connected by present bonds. For p > Pc zn infinite cluster is formed. Percolation theory (in a Bethe lattice approximation) was invented by Flory (1941) to describe gelation for three-functional polymers. 

In the previous section, we have briefly reviewed the basics of the lattice models of percolation where the systems have sites (or bonds) that are occupied or empty. In continuum percolation, the systems are composed of objects (or members) that are randomly placed in space. These objects may be of various sizes and shapes. If the latter are nonisotropic, one also considers the distribution of their orientations. Correspondingly, the values of the physical parameters that determine the bonding between two objects may vary from bond to bond, depending, say, on the local geometry and/or properties of the bond. 

A chain model described by the trajectory of a random walk on a lattice is called the lattice model (Figure 1.2(c)). The lattice constant a plays the role of the bond length. The simplest lattice model assumes that each step falls on the nearest neighboring lattice cell with equal probability [1], so that the connectivity function is given by... 

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