Percolation continuum

In the present case, one introduces conducting defects in the form of circles in two dimensions or spheres in three dimensions with randomly positioned centres having the possibility to overlap. The size of one defect is the characteristic length of the problem, as mentioned in the case of the fuse problem. 

We shall only consider the limit p very near to Pc when one can use the results of percolation. In this case, if the minimum possible distance between the clusters is 6min then, following the same argument as above, one gets or 

To conclude we recall that the formula (2.73) is still valid in the present case, and the values of the exponent 0 (= th here) are given in Table 2.2. 

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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... 

Recent studies [91-93] have shown that, for one component of the system undergoing thermoplastic deformation, the continuum percolation model can be used to predict the changes in the system with respect to a traditional pharmaceutical... 

The continuum percolation model predicts an excipient percolation threshold around 16% v/v. This can explain the important decrease in the critical point corresponding to the excipient percolation threshold, a critical point that governs the mechanical and release properties of the matrix. 

Millan, M., and Caraballo, I. (2006), Effect of drug particle size in ultrasound compacted tablets Continuum percolation model approach, Int. I. Pharrn., 310,168-174. 

Kuentz, M. T., and Leuenberger, H. (1998), Modified Young s modulus of microcrystalline cellulose tablets and the directed continuum percolation model, Pharm. Dev. Technol., 3(1), 1-7. 

Domes et al. (1987) reported hole mobilities of a benzotriazole derivative doped into a PC. The field dependencies were described as log t = pE. Domes et al. argued that the concentration dependence was consistent with predictions of the three-dimensional continuum percolation theory (Straley, 1982 Deutscher et al., 1983 Halperin et al., 1985 Feng et al., 1987). 

Fig. 1.5. (a) A typical link-element structure in the Swiss-cheese model of continuum percolation in two dimensions. The channel width is denoted by 6. (b) The dashed lines indicate the outline of the rectangular bond which approximates the narrow neck of a channel. 

In continuum percolation (see Section 1.2.1(g)), we suppose that the defects are introduced in a solid sample as randomly placed insulating holes with the shape of a circle (in two dimensions) or a sphere (in three dimensions) and we include the possibility of overlap of the defects (Swiss cheese model). This last possibility gives near Pc an infinite cluster with the the links having different cross-sectional width 6. This property is essentially responsible for the differences between lattice and continuum percolations. 

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. 

It may be mentioned here that this result is valid for continuum percolation as well, with the appropriate value (= Tf) for the exponent Tf. 

The scaling behaviour of the most probable fracture strength (jf, expressed by the fracture exponent Tf near the percolation threshold, has been investigated extensively. The theoretical results compare well with those observed experimentally, and in computer simulations. Although considerable progress has been made here, experimental results for continuum percolation are scarce, and more investigations are clearly necessary. 

Berkowitz and Ewing (1998) discuss continuum percolation, which differs from percolation on lattice networks. The important differences are ... 

The most important achievement of the work [53] was the implementation of SC-MC/RISM technique to probe the continuum percolation in the water subphase of Nafion. The solution of such a problem is of interest for... 

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. 

Finally, regarding the %c values we note that these were usually taken as an experimentally given parameter and very few attempts were made to account for their particular values in specific composites. Following the discussion in this chapter, we conclude that these are very sensitive to the dispersion of the particles, which is determined by their interaction during the fabrication of the composites. In particular, we note that the simple theories of continuum percolation were proper only under the assumption of a uniform dispersion of the particles and as such can serve only as indicators, or as giving bounds, for the %c values. Hence, for the determination of the values theoretically, or for the evaluation of these values experimentally, one needs to provide a description of the dispersion in a quantitative way. Attempts in this direction have begun only recently. 

Balberg, I. (1998) Limits onthe continuum-percolation transport exponents. Phys. Rev., B57, 13351. 

Meester, R. and Roy, R. (1996) Continuum Percolation, Cambridge University Press, Cambridge. 

Neda Z, Florian R and Brechet Y (1999) Reconsideration of continuum percolation of isotropically oriented sticks in three dimensions, Phys Rev E 59 3717-3719. 

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... 

Continuum percolation theory can be used to describe the effective properties of CLs, expressed through the volume fractions of components. For instance, the proton conductivity is given by... 

Kuhner G, Voll M (1993) Manufacture of carbon black. In Doimet J-B, Bansal RC, Wang M-J (eds) Carbon black science and technology. Taylor Francis, London, p 1 Kyrylyuk AV, van der School P (2008) Continuum percolation of carbon nanotubes in polymeric and colloidal media. Proc Nat Acad Sci USA 105 8221 Lacey D, Beattie HN, Mitchell GR, Pople JA (1998) Orientation effects in monodomain nematic liquid crystalline polysiloxane elastomers. J Mater Chem 8 53 Laird ED, Li CY (2013) Structure and morphology control in crystalline polymer-carbon nanotube nanocomposiles. Macromolecules 46 2877... 

Often RHJ, van der School P (2009) Continuum percolation of polydisperse nanofillers. Phys Rev Left 103 225704... 

For continuum percolation, the percolation probability is replaced by a percolation volume fraction, given by Xc = Pcf, where / is the filling factor (Hunt, 2005). For a polydisperse medium, Xc is a monotonically decreasing function of the polydispersity. 

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