Node percolation

Another typical problem of percolation theory is the problem of node percolation. A.s-suine that atoms of two kinds arc randomly placed in Ising s lattice (sec Figure 1.23c), where the black nodes are occupied by atoms with a permanent magnetic moment while the white ones are occupied by non-magnetic (diluent) atoms. 

However, the critical indices of the correlation length i/ and of the order parameter / in the percolation problem (equal for both bond and node percolation) take the values characteristic for the mean field approximation 0 = I and i/ = 0.5 for six-dimensional space d = 6 (cf. d = 4 for critical phenomena in Figure 2.44). 

Figure 3.58. Problem of bond and node percolation a model of gel formation during polymerization of a polyfunctional monomer in solution (Stanley et al., 1980 Efros, 1982 ) [Reprinted with permis.sion from Dynamics of Synergetic Systems Proc. Int. Symp. on Synergetics, Butlfeld, Germany, September 2f-29, 1979 Ed. M.Haken. Copyright by Springer-VerlagJ...

Figure 3.59. State diagram of the NP+IiMWIj system. DIG is Uio biiiodal curve of phase separation. C i.s the critical point. A is the line of the solution-gel transition, is the polymer concentration corre.sponding to the node percolation threshold l-he...

In random bond percolation, which is most widely used to describe gelation, monomers, occupy sites of a periodic lattice. The network formation is simulated by the formation of bonds (with a certain probability, p) between nearest neighbors of lattice sites, Fig. 7b. Since these bonds are randomly placed between the lattice nodes, intramolecular reactions are allowed. Other types of percolation are, for example, random site percolation (sites on a regular lattice are randomly occupied with a probability p) or random random percolation (also known as continuum percolation the sites do not form a periodic lattice but are distributed randomly throughout the percolation space). While the... 

Lymphatic fluid enters the node through several afferent lymphatics, percolates through most areas of the node and... 

Network structures are still determined by nodes and strands when long chains are crosslinked at random, but the segmental spacing between two consecutive crosslinks, along one chain, is not uniform in these systems which are currently described within the framework of bond percolation, considered within the mean field approximation. The percolation process is supposed to be developed on a Cayley tree [15, 16]. Polymer chains are considered as percolation units that will be linked to one another to form a gel. Chains bear chemical functions that can react with functions located on crosslinkers. The functionality of percolation units is determined by the mean number f of chemical functions per chain and the gelation (percolation) threshold is given by pc = (f-1)"1. The... 

In actual use for mobility control studies, the network might first be filled with oil and surfactant solution to give a porous medium with well-defined distributions of the fluids in the medium. This step can be performed according to well-developed procedures from network and percolation theory for nondispersion flow. The novel feature in the model, however, would be the presence of equations from single-capillary theory to describe the formation of lamellae at nodes where tubes of different radii meet and their subsequent flow, splitting at other pore throats, and destruction by film drainage. The result should be equations that meaningfully describe the droplet size population and flow rates as a function of pressure (both absolute and differential across the medium). 

In the above node-link-blob picture, the percolation cluster is self-similar up to a length scale in the sense that starting from the length scale the links contain blobs (and the dangling ends) which, in turn, are composed of links and blobs (and the dangling ends) up to the lowest scale (of the lattice). This self-similarity extends up to infinite scale at the percolation threshold (where becomes infinitely large). 

Several attempts have been made to relate these conductivity exponents tc or Sc with the percolation cluster statistical exponents discussed earlier. For example, in the node-link-blob model, the conductivity E of the network is proportional to the number of parallel links, while the... 

Using the node-link-blob model (see Section 1.2.1(d)) for the percolation cluster, one can in fact easily derive a rigorous bound of the fracture exponent Tf (Chakrabarti 1988, Ray and Chakrabarti 1988). This derivation is given below. One can see there that the above estimate for Tf in (3.12) turns out to be its lower bound. 

As discussed in Section 1.2.1, for such systems near the percolation threshold Pc the nearest-neighbour occupied bonds (or sites) form a statistically defined super-lattice , made of tortuous link-bonds (of chemical length Lc) crossing at nodes separated by an average distance the percolation correlation length (see Fig. 1.3 of Chapter 1). The external stress... 

In the node-link model of the Swiss-cheese system (see Section 1.2.1 (g)), each link has a distribution P 8) of channel width 6 and hence of local strength, as indicated in Fig. 1.5 of Chapter 1. One expects here the minimum value of = min to be of the order of L , where (Ap) e " is the chemical length of the singly connected link of the percolating network, and P 6) to be finite for 6 —> min-... 

TEM micrographs of blends made from 0.5% and 0.25% PANI-CSA in PMMA are shown in Fig. VI-8 (a and b) [61,279]. The PANI-CSA network can be seen quite clearly. Fig. VI-8 (a and b) resemble the typical scenario imagined for a percolating medium [277] with links (PANI-CSA fibrils), nodes (crossing points of the links) and blobs (dense, multiply connected regions). In the sample containing 0.5% PANI-CSA numerous weak links are clearly visible while for the 0.25% sample there are rather few links between the nodes and blobs. This indicates that at volume fractions below 0.5% PANI-CSA in PMMA, the network is unstable and tends to break up into disconnected blobs [285]. 

Because of problems with the FS approach, Stauffer (55) and de Gennes (56) advanced bond percolation as a description of polycondensation (see Figure 7). In the percolation model, bonds are formed at random between adjacent nodes on a regular or random d-dimensional lattice (57). In this approach, cyclic molecules are allowed and excluded volume effects are directly accounted for. 

The solution-gel transition is modelled by the percolation problem, which combines node and bond percolations. 

With a knowledge of the percolation threshold, the minimum necessary filling to give conducting composites can be predicted. For example, 9 for epoxysilicon resin filled with spherical particles of dispersed nickel is 0.25, and the critical parameter (X, determined by the number of bonds at conducting nodes in the lattice of the soli is 0.30. ... 

As it is known [115], the critical behaviour of the power of an infinite cluster (of the probability of a node belonging to this cluster) approaching the percolation threshold x is described by the scaling relationship ... 

The first extension of the percolation theory to address the problem of electrical transport in composite materials was conducted by Kirkpatrick rrsing a random resistor network model. Random resistor networks are created by assigning each node in the network with a random resistivity value and calculating the arrrent flow through the entire network at a fixed external voltage by solving Kirchoff s law at every node. Kirkpatrick s random resistor model provides a simple and convenient discrete model for the conductivity of a continuorrs medium if the spatial distribution of particles is known. As a... 

The theory of percolation helps to interpret this result. Let us consider a uniform network of points in space and let us introduce, progressively, bonds between nodes. At first isolated clusters of different sizes, of connected points are formed. When approaching the critical value p, called percolation threshold, one of these clusters grows very rapidly through the conglomeration of isolated clusters. For values greater than p, this cluster is infinite. The theory of percolation, applied to the study of conductivity, reproduces the equations of the effective medium, except at the immediate vicinity of the percolation threshold, where apparent conductivity then varies like ... 

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