Percolation lattice

We shall use the same machinery as that developed above in the case of the lattice fuse model. In order to mimic the insulator part, one takes capacitors with a given value Ci as insulating bonds. For the conducting bonds, one takes very large capacitor Cq such that Co Ci (Beale and Duxbury 1988). By this choice the voltage across a Cq capacitor is zero, as within the conductor the field must be zero, p is the fraction of Cq capacitors which are now seen as defects. Of course, the other possibility is to take 

The enhancement factor of the field near a long defect (made of n Cq capacitors) perpendicular to the electrodes is given by 

The probability to find a long defect made of n Cq capacitors is 

The distribution function F E) giving the probability that the sample will break when a field E is applied is obtained by a procedure identical to that we used for the fuse model. The result is 

In this limit, we consider clusters of conducting defects and their mean size is the percolation correlation length The voltage Vi between two such neighbouring clusters is given by Vi = V /L. The field between two clusters is increased as the field is zero inside a conducting cluster. The maximum value will be between the clusters with minimal distance between them i.e. one unit cell of the lattice. When the field between two such clusters 

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Alternatively, Leung and Eichinger [51] proposed a computer simulation approach which does not assume any lattice as the classical and percolation theory. Their simulations are more realistic than lattice percolation, since spatially closer groups form bonds first and more distant groups at later stages of network formation. However, the implicitly introduced diffusion control is somewhat obscure. The effects of intramolecular reactions were more realistically quantified, and the results agree quite well with experimental observations [52,53],... 

Lattice percolation models were the first spatial simulation models applied to the network build-up. Classic lattice or off-lattice percolation modeling is based on random introduction of bonds between components placed randomly on the lattice or in space [56-58]. They suffer from the rigidity of the system and disregard of conformational changes accompanying the structure growth. These assumptions implicitly mean that the bond formation is much faster than conformation changes. Such assumption is somewhat closer to reality for fast bond-... 

Bethe lattices, percolation, 39 10 Bethe tree model, 39 26 BET method, in heterogeneous catalysis, 17 15-17... 

Computer simulation in space takes into account spatial correlations of any range which result in Intramolecular reaction. The lattice percolation was mostly used. It was based on random connections of lattice points of rigid lattice. The main Interest was focused on the critical region at the gel point, l.e., on critical exponents and scaling laws between them. These exponents were found to differ from the so-called classical ones corresponding to Markovian systems irrespective of whether cycllzatlon was approximated by the spanning-tree... 

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). 

Quantitative analysis most probable failure current and distibution (a) Lattice percolation... 

In the very dilute limit p —> 1, there is clearly no difference between the two types of percolation. We thus expect, as above, a strong variation of 7f(p) near p = 1. In the dilute limit, when the defects are well separated but their number is not very small, the same argument of dangerous defect that we used in the case of lattice percolation can again be applied. Now L is the ratio between the size of the sample and the diameter of one defect. All the results we established above are valid here. 

The principal results are that the exponents of the failure current are higher than those of discrete percolation and that the failure voltages have a completely different behaviour. While in the present case of continuum percolation, they always go to zero at Pc, for lattice percolation the failure voltage either reaches a finite value (in two dimensions) or even diverges (in three dimensions). This just reminds us that the physical quantity which brings the failure is the local current density. 

This formula is valid in all cases, with the exponent (j) (= z/ for lattice percolation) dependent on the dimension and on the type of percolation (see below the results for continuum percolation). 

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. 

Percolation Lattices. Percolation theory requires that space be represented as a lattice, often infinite in extent. In Figure 4.19, for example, two-dimensional space is discretized on to a square lattice. The points of... 

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. 

In this chapter, the emphasis will be then on trying to understand the a x) dependence as observed in those composites in view of the recent extensions of classical lattice percolation theory to systems in the continuum. The basic characteristic of o x) that is common to all composite materials is a sharp rise in cr as x increases. This rise is followed by a monotonic moderate rise in the o x) dependence. In order to get the desired understanding of this behavior, the first part of this chapter will provide the background to percolation theory [1-4] and the tools required for the discussion of the cr(x) dependence, while the other part will be devoted to a comparison of the experimental observations of ours and others to the expectations that follow percolation theory. In particular, we will try to understand the multiple sharp... 

It has been possible to directly image the percolation network at the surface of a CB-polymer composite. An early report is that of Viswanathan and Heaney [24] on CB in HOPE in which it was shown that there are three regions of conductivity as a function of the length L, used as a metric for the image analysis. Below I = 0.6pm, the fractal dimension D of the CB aggregates is 1.9 0.1. Between 0.8 and 2 pm, the data exhibit D = 2.6 0.1 while above 3 pm, D = 3 corresponding to homogeneous behavior. Theory predicts D = 2.53. It is not obvious that the carbon black-polymer system should be explainable in terms of standard percolation theory, or that it should be in the same universality class as three-dimensional lattice percolation problems [24]. Subsequent experiments of this kind were made by Carmona [25, 26]. 

Abstract. The square and cubic lattice percolation problem and the selfavoiding random walk model were simulated by Monte Carlo method in order to obtain new understanding of the fractal properties of branched and hnear polymer molecules. The central point of this work refers to the comparison between the cluster properties as they emerge from the percolation problem on one hand and the random walk properties on the other hand. It is shown that in both models there is a drastic difference between two and three dimensional systems. In three dimensions it is possible to find a regime where the properties converge towards simple non-avoided random walk, while in two dimensions the topological reasons prevent a smooth transition of the properties pertaining to avoided and non-avoided random walks. 

The main topics in lattice theories, which are relevant for the polymer subject are avoided random walk, lattice percolation [3] and lattice spin models. In this work we shall put the emphasis on the numerical investigation of the systems in the framework of lattice percolation methodologies and avoided random walks on square and cubic lattices. 

Percolation phenomena deal with the effect of clustering and coimectivity of microscopic elements in a disordered medium [129], Percolation theory represents a random composite material as a network or lattice structure of two or more distinct types of microscopic elements or phase domains, the so-called percolation sites. These elements represent mutually exclusive physical properties, e.g., electrically conducting vs. isolating phase domains, pore space vs. solid matrix, atoms with spin up vs. spin down states. Here, we will refer to black and white elements for definiteness. The network onto which black and white elements of the composite medium are distributed could be continuous (continuum percolation) or discrete (discrete or lattice percolation) it could be a disordered or regular network. With a probability p a randomly chosen percolation site will be... 

For lattice percolation, two types of problems are distinguished. For the hitherto described site percolation problems, clusters are formed by white or black sites of the lattice. For bond percolation, the same statistical concept is applied to the connections or bonds between lattice sites with probability / , a randomly chosen bond will be occupied by a white element and otherwise, that is, with the complementary probability (1 — / ), the bond will remain black. Such a system is depicted in Figure 3.43a. 

Other physical properties like correlation lengths and percolation probabilities follow as well power laws in the vicinity of pc, however with different critical exponents. Values of the critical exponent /r in Equation 3.102 are known in 2D and 3D from computer simulations (Isichenko, 1992). For lattice percolation in 2D, it is u. 1.3... 

The now popular concept of percolation has proved quite successful in many fields in which a macroscopic property depends on the existence of a connected path within a two-phase discrete medium, most often a regular 2D or 3D arrangement of sites (site lattice percolation). Typical features related to percolation are the existence of a critical phenomenon, for instance, a threshold concentration of conducting sites when conduction is considered, and of a power law dependence with respect to the critical quantity in the close vicinity of the critical point. Both the site percolation thresholds and the power exponents have well-established theoretical values for any given lattice geometry and coimection rules [193],... 

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