Percolation theory continuum

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). 

In the rest of this chapter, we will discuss briefly the theoretical ideas and the models employed for the study of failure of disordered solids, and other dynamical systems. In particular, we give a very brief summary of the percolation theory and the models (both lattice and continuum). The various lattice statistical exponents and the (fractal) dimensions are introduced here. We then give brief introduction to the concept of stress concentration around a sharp edge of a void or impurity cluster in a stressed solid. The concept is then extended to derive the extreme statistics of failure of randomly disordered solids. Here, we also discuss the competition between the percolation and the extreme statistics in determining the breakdown statistics of disordered solids. Finally, we discuss the self-organised criticality and some models showing such critical behaviour. 

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... 

This chapter studies the local and global structures of polymer networks. For the local structure, we focus on the internal structure of cross-Unk junctions, and study how they affect the sol-gel transition. For the global structure, we focus on the topological connectivity of the network, such as cycle ranks, elastically effective chains, etc., and study how they affect the elastic properties of the networks. We then move to the self-similarity of the structures near the gel point, and derive some important scaling laws on the basis of percolation theory. Finally, we refer to the percolation in continuum media, focusing on the coexistence of gelation and phase separation in spherical coUoid particles interacting with the adhesive square well potential. 

Percolation models are roughly classified into percolation on regular lattices and percolation in continuum space. Both derive the scaling laws near the percolation threshold by focusing on the self-similarity of the connected objects. The percolation theory is suitable for the study of fluctuations in the critical region, but has a weak point in that the analytical description of the physical quantities in wider regions is difficult. 

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... 

Figure 6.12 Experimental co-operative behavior of PEDOT PSS latex particles and HiPCO SWCNTs (circles). Pitting of data using a multi-component continuum connectedness percolation theory (line). ...

An investigation into the co-operative behavior demonstrates that the percolation threshold can be modeled using a multi-component continuum connectedness percolation theory. A deeper investigation into the co-operative nature of the two conductive components revealed that the contribution of the SWCNTs to the overall composite conductivity is minimal, and that the role of the SWCNTs is more morphological and likely to be that of a kind of template or scaffold for the deposition of a connected PEDOT PSS phase. 

For completeness, note that several researchers have exploited the well-developed analytical theories of the stmcture of fluids to model percolation in mixtures of interacting particles. By proposing various extensions of the multicomponent Omstein-Zernike equation, coupled with connectivity definitions from continuum percolation theories, simplified analytical expressions are derived for the percolation threshold of a composite system subjected to interparticle and medium-induced interactions. However, to date, simulations dominate the study of dynamic percolation. 

Over the last decade, there has been considerable interest in the development of off-lattice models and theories describing percolation phenomena. Interest in percolation concepts has been spurred by the rather wide variety of applications for which such ideas are thought to be useful. These applications include the electrical conductivity and dielectric properties or permeability of composite materials, gelation, analysis of hydrogen bond networks, and reactions in porous catalysts. Recent progress in the development of off-lattice or, as they are most often called, continuum models of percolation began with the work of Coniglio et and Haan... 

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. 

Another deviation from mixture theory occurs in the manner with which the local pressure is born by each of the components. Standard mixture theory states that all fields other than the velocity field are related linearly by their concentrations in the mixture. Gray and Thornton suggested, however, that pressure was rather divided in such a way as to reflect the likely manifestation of kinetic sieving in a continuum formulation. They suggested that, while the small particles move downward, or percolate, through the matrix, they bear less of the overburden pressure than their concentration would suggest, and the large particles carry more of the load. In other words, the partial pressures may not be proportional to their concentrations, that is, if the partial pressure coefficients are defined as... 

Juarez-Maldonado, R., Chdvez-Rojo, M. A., Ramtrez-Gonzdlez, P. E., Yeomans-Reyna, L., and Medina-Noyola, M. 2007. Simplified self-consistent theory of colloid dynamics. Phys. Rev. E 76 062502. Sahimi, M. 1993. Flow phenomena in rocks From continuum models to fractals, percolation, cellular automata, and simulating annealing. Rev. Mod. Phys. 65 1393. 

The excluded volume theory is the most commonly used continuum analytical model of percolation. The excluded volume of an object is defined as the volume around the object into which another identical object cannot enter without contacting the first object as illustrated in Figure 2. ° The principal concept in the excluded volume model is that the percolation threshold of a system is determined by the excluded volume of filler particles, rather than their true volume. This is particularly applicable to asymmetrical, unaligned objects for which the excluded volume can differ significantly from their true volume. Therefore, this model has been applied to describe critical percolation phenomena for a wide variety of filler geometries. In addition, excluded volume arguments provide useful theoretical approximations in many computational stu-dies. Excluded volume solutions were first formulated for soft-core (interpenetrable) fillers, and later extended to core-shell (impenetrable hard-core surrounded by a penetrable shell) fillers. ... 

Specifically for gelation, we will discuss in Sect. C.V. various modifications of the simple percolation model of Fig. 1 and check if the exponents diange. In most cases, they do not in particular, the lattice structure (simple cubic, bcc, fee, spinels ) is not an important parameter since different lattices of the same dimensionality d give the same exponents within narrow error bars. More importantly, percolation on a continuum without any underlying lattice structure has in two and three dimensions the same exponents, within the error bars, as lattice percolation. In the classical Flory-Stockmayer theory which does not employ any periodic lattice structure, the critical exponents are completely independent of the functionality f of the monomers or the space dimensionality d. But if the system is not isotropic or if the gel point is coupled with the consolute point of the binary mixture solvent-monomers , the exponents may change as discussed in Sect. D. 

We have already mentioned that the lattice structure, while used for most percolation studies, is not really necessary and that even without the help of a lattice the critical exponents seem to have invariable lattice values. According to the simple classical theory this is not the case since die radius of trees on a periodic lattice (with excluded volume effects) increase for large cluster masses s at least with s (in d dimensions) whereas in the classical theory on a continuum a Caley tree has a radius varying asymptotically with s, independent of d. 

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