Scaling theory of percolation

D. Stauffer. Scaling theory of percolating clusters. Phys Rep 54 1-74, 1979. 

Stauffer, D., Scaling theory of percolation clusters. Physics Reports, 1979, 54, 1-74. 

Spiegelman, S. (1971). Quart. Rev. Biophys., 4, 215-Srejder, Yu. (1965). Monte Carlo methods. Fizmatgiz, Moscow (in Russian). Stauffer, D. (1979). Scaling theory of percolation clusters. Phys. Rev., 54, 1-74. Steeb, W. H. (1978). The Lie derivative, invariance conditions, and physical laws, Z. Naturforsch., 33a, 742-8. 

This chapter reviews some of the work done on disordered carbon-black-polymer composites by my collaborators and myself over the past several years. These composites have widespread commercial applications. A qualitative analysis of a transmission electron microscope image is presented. Quantitative analyses of scanning probe microscope images and dc electrical resistivity data are presented. The resistivity and linear expansion of a typical composite between 25 and 180 C are measured and analyzed. The scaling theory of percolation provides a good explanation of most of our data. 

Disordered carbon-black-polymer composites have many common uses in modem technology. These uses include inks, automobile tires, reinforced plastics, wire and cable sheaths, antistatic shielding, resettable fuses, and self-regulating heaters. As an immediate example, the inked letters on this page consist of a disordered carbon-black-polymer composite bound to the surface of the paper. Despite these widespread applications, many of the important physical properties of these composites are not well understood. There is a long history of experimental and theoretical work on disordered carbon-black-polymer composites [7]. One of the most exciting recent advances in the field has been the application of the scaling theory of percolation [2] to these composites. This is based on the many similarities between the percolative transition in a disordered conductor-insulator composite and the thermodynamic phase transition common in many materials. 

The scaling theory of percolation predicts that the correlation length will obey a scaling law of form [2]... 

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). 

D.Stauffer. Scaling Theory of Percolation Clusters. Phys.Rep.54 1(1979)... 

A more detailed relationship between conductivity and volume fraction of PANI-CSA in PMMA is shown at low volume fractions in the inset to Fig. VI-9. In order to identify the percolation threshold more precisely, the data were fitted to the scaling law of percolation theory [277],... 

The power laws for viscoelastic spectra near the gel point presumably arise from the fractal scaling properties of gel clusters. Adolf and Martin (1990) have attempted to derive a value for the scaling exponent n from the universal scaling properties of percolation fractal aggregates near the gel point. Using Rouse theory for the dependence of the relaxation time on cluster molecular weight, they obtain n = D/ 2- - Df ) = 2/3, where Df = 2.5 is the fractal dimensionality of the clusters (see Table 5-1), and D = 3 is the dimensionality of... 

In order to identify the percolation threshold more precisely, the data were fit to the scaling law of percolation theory [198],... 

The scaling of the relaxation modulus G(t) with time (Eq. 1-1) at the LST was first detected experimentally [5-7]. Subsequently, dynamic scaling based on percolation theory used the relation between diffusion coefficient and longest relaxation time of a single cluster to calculate a relaxation time spectrum for the sum of all clusters [39], This resulted in the same scaling relation for G(t) with an exponent n following Eq. 1-14. 

The classical theory predicts values for the dynamic exponents of s = 0 and z = 3. Since s = 0, the viscosity diverges at most logarithmically at the gel point. Using Eq. 1-14, a relaxation exponent of n = 1 can be attributed to classical theory [34], Dynamic scaling based on percolation theory [34,40] does not yield unique results for the dynamic exponents as it does for the static exponents. Several models can be found that result in different values for n, s and z. These models use either Rouse and Zimm limits of hydrodynamic interactions or Electrical Network analogies. The following values were reported [34,39] (Rouse, no hydrodynamic interactions) n = 0.66, s = 1.35, and z = 2.7, (Zimm, hydrodynamic interactions accounted for) n = 1, s = 0, and z = 2.7, and (Electrical Network) n = 0.71, s = 0.75 and z = 1.94. 

At the mesoscopic scale, interactions between molecular components in membranes and catalyst layers control the self-organization into nanophase-segregated media, structural correlations, and adhesion properties of phase domains. Such complex processes can be studied by various theoretical tools and simulation techniques (e.g., by coarse-grained molecular dynamics simulations). Complex morphologies of the emerging media can be related to effective physicochemical properties that characterize transport and reaction at the macroscopic scale, using concepts from the theory of random heterogeneous media and percolation theory. 

S. Greenspoon Finite-size effects in one-dimensional percolation a verification of scaling theory. Canadian J. Phys. 57, 550-552 (1979)... 

In this study, we have shown how gas-liquid flow through a random packing may be represented by a percolation process. The main concepts of percolation theory allow us to account for the random nature of the packing and to derive a theoretical expression of the liquid flow distribution at the bed scale. This flow distribution allows us to establish an averaging formula between the particle and bed scales. Using this formula, we propose the bed scale modelling of some transport processes previously modelled at the particle scale. 

For a quantitative analysis of the scaling and cross-over behavior of the a.c.-conductivity above the percolation threshold we refer to the predictions of percolation theory [128, 136, 137] ... 

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... 

It should be noted that, on one hand, an approach such as this is sufficiently closely related to the fluctuation theory of disperse systems developed in Shishkin s works [73], and on the other hand, it reduces to one of the variants of the flow problems in the percolation theory [78, 79] according to which the probability of the existence of an infinite liquid-like cluster depends on the value of the difference (P — Pcr), where Pcr is the flow threshold. At P < Pcr, only liquid-like clusters of finite dimensions exist which ensure the glassy state of liquid. It is assumed that at P > Pcr and (P — Pcr) 1 the flow probability is of the following scaling form ... 

The idea that star formation can have an induced component which depends on the available stellar population in the disk was first realized by Elmegreen and Lada and Herbst and Assousa, and at about the same time by Mueller and Amett for steUar systems. We shall discuss the effect of such an assumption on models of star formation in local models in Section IV now we can show that this picture carries over quite nicely into the global models. However, in these pictures, since there has been little analytic work on the large-scale structure (with the exception of Shore and Fujimoto and Ikeuchi ), we must confine the discussion to the more abstract aspects of percolation theory on differentially rotating planes and then discuss the reinterpretation of the results in light of the modeling that has been done to date. 

We now define some statistical quantities of interest in percolation theory. Let ns p) denote the number of clusters (per lattice site) of size s. In fact, a detailed knowledge of ns p) would give us a lot of information on the percolation statistics, as most of the quantities of interest can be extracted from various moments of the cluster size distribution n. Although, in general, we do not have any analytic knowledge about this distribution function ns p) near Pc, we can utilise the powerful observation of scaling behaviour of ns p) near Pc (see the next section). 

Scaling Exponents for Classical and Percolation Theories of Gelation... 

Anomalous subdiffusion occurs on percolation clusters or on objects that in a statistical sense can be described as fractal, by which we mean that selfsimilarity describes simply the scaling of mass with length. Connections between v, the fractal dimension of the cluster, D, and the spectral dimension, d, have been established, relations that were originally derived by Alexander and Orbach [35], who developed a theory of vibrational excitations on fractal objects which they called fractons. An elegant scaling argument by Rammal and Toulouse [140] also leads to these relations, and we summarize their results. 

Luxmoore, R.J., G.V. Wilson, P.M. Jardine, and R.H, Gardner. 1990. Use of percolation theory and latin hypercube sampling in field-scale solute transport investigations, p. 437-439. In D.-Q. Lin (ed.) Proc. 1st Int. Symp. For. Soils, Harbin, People s Republic of China. 22-27 July. Northeast Forestry Univ., Harbin, People s Republic of China. 

Table 2.3. Scaling exponents for classical and percolation theories of gelation ...

The reader is referred to recent reviews" of percolation theory for z = 1 for a more complete study. Here we summarize some important results, which we expect to carry over to the environmental percolation problem with z. oxp>p and I> z P) assumed to have a scaling form,... 

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. 

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