Percolation theory critical exponents

It was shown in the preceding sections that most three-dimensional versions of the percolation theory have the same critical exponents. They differ only if phase separation occurs during the percolation process. Thus, if gelation can be described by the percolation theory, the exponent values must be independent of the chemical system studied. 

In other words, the random polycondensates have a critical exponent of yc = 1 while the ABC type under restriction has an exponent of yc = 2. Percolation theory yields Yc — 1.843,44). It has often been stated that the FS theory is characterized by a mean-field critical exponent of yc = 1, but now we see that the FS theory is more flexible and shows critical exponents between 1 and 2. 

If the estimated fitting parameters are compared to the predicted values of percolation theory, one finds that all three exponents are much larger than expected. The value of the conductivity exponent ji=7A is in line with the data obtained in Sect. 3.3.2, confirming the non-universal percolation behavior of the conductivity of carbon black filled rubber composites. However, the values of the critical exponents q=m= 10.1 also seem to be influenced by the same mechanism, i.e., the superimposed kinetic aggregation process considered above (Eq. 16). This is not surprising, since both characteristic time scales of the system depend on the diffusion of the charge carriers characterized by the conductivity. 

According to Stauffer (1979), A complete understanding of percolation would require [one] to calculate these exponents exactly and rigorously. This aim has not yet been accomplished, even in general for other phase transitions. The aim of a scaling theory as reviewed here is more modest than complete understanding We want merely to derive relations between critical exponents. Three principal methods currently employed to derive critical exponents are (i) series expansions, (ii) Monte Carlo simulation, and... 

Table III summarizes results for the critical exponents and critical coverages, obtained by percolation theory analysis of the protonic conduction processes for lysozyme and purple membrane, and compares these values with theory.

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. 

Theoretical and experimental treatments of gels go hand-in-hand. The former are covered first because they will help us understand gel point and other concepts. Two main theories have been used to interpret results of experimental studies on gels the classical theory based on branching models developed developed by Floiy and Stockmayer, and the percolation model credited to de Gennes. Gelation theories predict a critical point at which an infinite cluster first appears. As with other critical points, the sol-gel transition can be in general characterized in terms of a set of generally applicable (universal) critical exponents. 

Also similar to percolation is the existence of a critical exponent in the mode-coupling theory the longest relaxation time r scales as... 

In the classical theory, however, the neglect of loops significantly affects the size distribution and other properties of the clusters as one approaches the gel point. Some of the critical exponents that describe these properties in the classical theory and in percolation theory near p Pc are compiled in Table 5-1 (Martin and Adolf 1991). 

To test this theory, the room temperature conductivity of "Nafion" perfluorinated resins was measured as a function of electrolyte uptake by a standard a.c. technique for liquid electrolytes (15). The data obey the percolation prediction very well. Figure 9 is a log-log plot of the measured conductivity against the excell volume fraction of electrolyte (c-c ). The principal experimental uncertainty was in the determination of c as shown by the horizontal error bars. The dashed line is a non-linear least square law to the data points. The best fit value for the threshold c is 10% which is less than the ideal value of 15% for a completely random system. This observation is consistent with a bimodal cluster distribution required by the cluster-network model. In accord with the theoretical prediction, the critical exponent n as determined from the slope of... 

The frequency dependence of the dielectric constant for different levels of conductivities can also be analysed with regard to percolation theory. In fact this theory developed by Stauffer [141] shows that percolation aggregates can be described with seven critical exponents of power laws of (p — Pc) where p and pc are respectively the concentration in inclusions and the critical concentration at percolation threshold. It is shown that the seven critical exponents are linked through five scaling laws having only two exponents are independent variables to be fitted by experiment and not predicted by scaling theory. 

Some works have also been carried out for linking the observed frequency dependence to critical exponents of percolation theory. 

The field theory for SAWs on the percolation cluster developed in Ref. [22] supports an upper critical dimension d p = 6. The calculation of Up was presented to the first order of perturbation theory, however the numerical estimates obtained from this result are in poor agreement with the numbers observed by other means. In particular, they lead to estimate that i/p i/ in d = 3. Recently this investigation has been extended to the second order in perturbation theory [101], which leads to the qualitative estimates of critical exponents in good agreement with numerical studies and Flory-like theories. 

The critical exponent v depends only on the dimensionality D, and that feature is known as universality. The well-known values of v in percolation theory are 1.33 for D = 2 and 0.88 for D = 3. 

As Balberg notes in a review The electrical data were explained for many years within the framework of interparticle tunneling conduction and/or the framework of classical percolation theory. However, these two basic ingredients for the understanding of the system are not compatible with each other conceptually, and their simple combination does not provide an explanation for the diversity of experimental results [17]. He proposes a model to explain the apparent dependence of percolation threshold critical resistivity exponent on structure of various carbon black composites. This model is testable against predictions of electrical noise spectra for various formulations of CB in polymers and gives a satisfactory fit [16]. 

Equation (2b) is a scaling law depicting the conductive behavior in the vicinity of the percolation threshold, the value of the critical exponent y being 1.6 to within 0.2. Equation (2c) expresses the composite conductivity dependence upon conductor concentration beyond the percolation threshold. Equation (2c) is a simplified form, valid in the case of conductor-insulator mixtures, of a more general equation derived in different ways by Bruggeman (70), Bottcher (71) and Landauer (72) and known as the Effective Medium Theory, (E.M.T.), formula ... 

The percolation takes place if the critical volume fraction of secondary nanotube agglomerates Vy ggg is reached. According to classical percolation theory, the conductivity increase can be described with power law behavior (Eq. 5.9) with cr the plateau value of conductivity and the critical exponent (see also Appendix). 

Vojta and Sknepnek also performed analogous calculations for the quantum percolation transition at p = pp, J < 0.16/ and the multicritical point 2itp=pp,J = 0.16/. A summary of the critical exponents for all three transitions is found in Table 3. The results for the percolation transition are in reasonable agreement with theoretical predictions of a recent general scaling theory of percolation quantum phase transitions P/v = 5/48, y/v = 59/16 and a dynamical exponent oi z = Df = (coinciding with the fractal dimension of the critical percolation cluster). 

Sokolov, I. M. (1986). Dimensions and other Geometrical Critical Exponents in Percolation Theory. Uspekhi FizicheshikhNauk, 150(2), 221-256. 

To explain the power law evolution of the elastic modulus and the exponent a close to 4, it has been proposed that an analogy could exist between a gel and a percolation cluster (Stauffer, 1976). In this theory, elastic properties are expected to scale as Eot P— PcY where P is the probability for a site to be occupied (or a bond to be created) and Pc is the percolation threshold (defined as the magnitude of P above which an infinite cluster exists). Analytical work and simulations have calculated a critical exponent characteristic of the elastic modulus) and r is close to 4 (Feng, 1984 Kan tor, 1984). 

Critical Exponents for Near-Threshold Scaling Behavior in Percolation Theory. 

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