Percolation theory definition

Although percolation theory deals with random systems, modeling and numerical calculations for percolation are usually carried out for a lattice of some definite geometry. To reach conclusions which do not depend on the details of lattice geometry but on dimensionality only, and thus are valid for the random system of interest, some invariant quantities must be constructed. 

Table 3.5 summarises the main definitions from percolation theory and their interpretation in terms of bubbles in the adherence region. The main idea is,... 

In materials of infinite extent, the above definitions remain valid. As noted previously (Chapter 4), for pore space topologies with a given coordination number, there exists a critical filling probability (porosity). In materials with filling probabilities above this critical value, the size of the largest cluster is comparable to the size of the lattice. The presence of this lattice spanning cluster does not require that the material be finite in extent in fact, most analytical results in percolation theory assume that the lattice is infinite. For... 

It should be noted that phase inversion prediction models focus on only a single composition, whereas in reality, co-continuous structures are observed over a composition range. Considering the definition of co-cmitinuous structure and equations based on the percolation theory, a model was proposed to correlate a continuity index (/) with the volume fraction at onset of co-continuity (0,- ) (see Table 7.3) (Lyngaae-Jorgensen et al. 1999). Numerical simulation predicted cr to be about 0.2 for classical percolation in three-dimensional systems (Dietrich and Airmon 1994 Potschke and Paul 2003). 

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... 

The polymers physical aging represents itself the structure and properties change in time and is the reflection of the indicated materials thermodynamically nonequilibriiun nature [61, 62], As a rule, the physical aging results to polymer materials brittleness enhancement and therefore, the ability of structural characteristics in due course prediction is important for the period of estimation of pol5mier products safe exploitation. For cross-linked polymers the quantitative estimation of structure and properties changes in physical aging process was conducted in Refs. [63, 64] within the frameworks of fracture analysis [65] and cluster model of polymers amorphous state structure [7, 66]. The authors of Ref. [67] use the indicated theoretical models for the description of PC physical aging. Besides, for PC behavior closer definition in the indicated process such theoretical notions were drawn as structure quasiequilibrium state [68] and the thermal cluster model [69], which is one from variants of percolation theory. 

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. 

Aoki also considers the stochastic aspects of phase propagation mechanism and relates his analysis to the theory of percolation and the fractal dimension of the system. In this approach the Nemst equation for charge transfer at the substrate/film interface is used to compute the probability of the presence of a conductive seed or nucleus. When the potential is incremented, this seed can then grow in a one-dimensional manner governed by the propagation rate constant kp or the kinetic parameter to form a conductive pillar of a definite length. New nuclei can also form at the support electrode/film interface during the potential... 

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