Percolation theory critical threshold

Figure 2.9.3 shows typical maps [31] recorded with proton spin density diffusometry in a model object fabricated based on a computer generated percolation cluster (for descriptions of the so-called percolation theory see Refs. [6, 32, 33]).The pore space model is a two-dimensional site percolation cluster sites on a square lattice were occupied with a probability p (also called porosity ). Neighboring occupied sites are thought to be connected by a pore. With increasing p, clusters of neighboring occupied sites, that is pore networks, begin to form. At a critical probability pc, the so-called percolation threshold, an infinite cluster appears. On a finite system, the infinite cluster connects opposite sides of the lattice, so that transport across the pore network becomes possible. For two-dimensional site percolation clusters on a square lattice, pc was numerically found to be 0.592746 [6]. 

The main conclusion of the percolation theory is that there exists a critical concentration of the conductive fraction (percolation threshold, c0), below which the ion (charge) transport is very difficult because of a lack of pathways between conductive islands. Above and near the threshold, the conductivity can be expressed as ... 

The initial porous texture of a catalyst pellet and the change in texture caused by metal deposition in it can be described using the percolation theory. In the percolation approach the pellet is constructed as a binary interdispersion of void space and (deposited) solid material. In this binary interdispersion, the void space can exist as (1) isolated clusters surrounded by solid material or (2) sample overspanning void space that allows mass transport from one side to the other. The total void space c can be split into the sum of the volume fraction of isolated clusters t1 and the volume fraction of accessible void space tA, If is below a critical value, called the percolation threshold all the void space is distributed as isolated clusters and transport is impossible through the pellet. 

The electrical percolation behavior for a series of carbon black filled rubbers is depicted in Fig. 26 and Fig. 27. The inserted solid lines are least square fits to the predicted critical behavior of percolation theory, where only the filled symbols are considered that are assumed to lie above the percolation threshold. According to percolation theory, the d.c.-conductivity Odc increases with the net concentration 0-0c of carbon black according to a power law [6,128] ... 

An explanation of the observed relaxation transition of the permittivity in carbon black filled composites above the percolation threshold is again provided by percolation theory. Two different polarization mechanisms can be considered (i) polarization of the filler clusters that are assumed to be located in a non polar medium, and (ii) polarization of the polymer matrix between conducting filler clusters. Both concepts predict a critical behavior of the characteristic frequency R similar to Eq. (18). In case (i) it holds that R= , since both transitions are related to the diffusion behavior of the charge carriers on fractal clusters and are controlled by the correlation length of the clusters. Hence, R corresponds to the anomalous diffusion transition, i.e., the cross-over frequency of the conductivity as observed in Fig. 30a. In case (ii), also referred to as random resistor-capacitor model, the polarization transition is affected by the polarization behavior of the polymer matrix and it holds that [128, 136,137]... 

Recently percolation theory is starting to be applied to the study of hydrophilic matrix systems. Figure 41 shows an example of the changes observed in several release parameters employed to estimate the critical point and the related percolation threshold in hydrophilic matrices prepared using KC1 as the model drug [73],... 

Application of the percolation theory allows explanation of the changes in the release and hydration kinetics of swellable matrix-type controlled delivery systems. According to this theory, the critical points observed in dissolution and water uptake studies can be attributed to the excipient percolation threshold. Knowledge of these thresholds is important in order to optimize the design of swellable matrix tablets. Above the excipient percolation threshold an infinite cluster of this component is formed which is able to control the hydration and release rate. Below this threshold the excipient does not percolate the system and drug release is not controlled. 

As percolation theory predicts, the studied properties show a critical behavior as a function of the volumetric fraction of the components. A critical point has been found between 21 and 26% v/v of excipient plus initial porosity (see Table 24). This critical point can be attributed to the excipient percolation threshold. 

Therefore, the results obtained from the kinetics analysis are in agreement with the release profiles, indicating a clear change in the release rate and mechanism between matrices containing 90 and 95% w/w of drug (5-10% w/w of excipient). The existence of a critical point can be attributed to the excipient percolation threshold. From the point of view of percolation theory, this means that above 10% w/w of FIPMC K4M, the existence of a network of HPMC (able to form a hydrated layer from the first moment) controls the drug release. 

Careri et al. (1986), using the framework of percolation theory, analyzed the explosive growth of the capacitance with increasing hydration above a critical water content (Fig. 14). The threshold for onset of the dielectric response was found to he 0.15 h for free lysozyme and 0.23 h for the lysozyme—substrate complex. In the percolation model the thresh-... 

The following simple experiment demonstrates the essential concept of the critical threshold for percolation. A mixture of small plastic and metal balls of equal size is poured into a beaker with a crumpled-foil electrode at the bottom, another crumpled-foil electrode is pressed onto the top, and the electrodes are connected to a battery through an ammeter. Current is measured as a function of the composition (i.e., fraction of metal balls) of the conductor/insulator mixture. There is a critical composition below which no current flows and above which the conductivity increases nearly exponentially. At this threshold the two electrodes suddenly become spatially connected along a statistical pathway originating in the random medium. Percolation theory tells us that the critical composition is 0.25 fraction metal balls, a remarkably low concentration. This is perhaps not an intuitive result. 

The percolation probability (q) for the lattice models is defined as the probability that a given site (or bond) belongs to an infinite open cluster (47). It is fundamental to percolation theory that there exists a critical value qc of q such that 9(q) = 0 3t q < qc, and (q) > 0 if > qc. The value qc is called the critical probability or the percolation threshold. Mathematical methods of calculating this threshold are so far restricted to two dimensions, consistent with the experience in the field of phase transitions that three-dimensional problems in general cannot be solved exactly (12,13). Almost all quantitative information available on the percolation properties of specific lattices has come from Monte Carlo calculations on finite specimens (8,11,12). In particular. Table I summarizes exactly and approximately known percolation thresholds for the most important two- and three-dimensional lattices. For the bond problem, the data presented in Table I support the following well-known empirical invariant (8)... 

The central result of percolation theory is the existence of the critical probability pc, the percolation threshold at which a cluster of infinite size appears. In a one-dimensional lattice, the percolation threshold is obviously equal to one. For higher dimensions, pc will be smaller than one. To illustrate this central result, we consider the Bethe lattice (also called the Cayley tree). 

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. 

Crete surface to the bulk of the concrete. Permeability is high (Figure 1.6) and transport processes like, e. g., capillary suction of (chloride-containing) water can take place rapidly. With decreasing porosity the capillary pore system loses its connectivity, thus transport processes are controlled by the small gel pores. As a result, water and chlorides will penetrate only a short distance into concrete. This influence of structure (geometry) on transport properties can be described with the percolation theory [8] below a critical porosity, p, the percolation threshold, the capillary pore system is not interconnected (only finite clusters are present) above p the capillary pore system is continuous (infinite clusters). The percolation theory has been used to design numerical experiments and apphed to transport processes in cement paste and mortars [9]. 

Percolation theory applied to the two-phase system It is a well known fact that the transition of modulus takes place in the two-phase system from a rubbery state to a glassy state at the critical region (not a point) as a function of the composition. Morphologically the transition is understood as the reverse of phases. Recently by introducing the concept of percolation concept, the transition composition is defined by the elastic percolation threshold. The scaling rule proposed by de Gennes [9] is applied to such a variation of modulus, using the critical composition at the elastic percolation threshold. 

The biodegradability of starch in the plastic matrix mainly depends on the accessibility of starch to microbes and on the coimectivity of starch particles each other. Wool and Cole (8) described a simulation model based on percolation theory for predicting accessibility of starch in LDPE to microbial attack and add hydrolysis. This model predicted a percolation threshold at 30% (v/v) starch irrespective of component geometry and other influential factors. Two critical aspects, the bioavailability and the kinetics of the starch hydrolysis in the plastic matrix must be examined before such blends could be applied as controlled release formulation of pestiddes. The goal of this work was to develop a kinetic model describing the degradation and release of starch blended with hydrophobic plastics. 

An important quantity in percolation theory is the percolation probability P p), which gives the probability that given site belongs to the infinite cluster. One can show that there exists a critical value Pc (also called the percolation threshold) such that P(p) is... 

We have then that the hopping model, while frequently called the percolation model [6], is essentially a percolation threshold model that is not associated a priori with the critical behavior of the conductivity as in percolation theory (Eq. (5.6)). In passing we further note that in the limiting case of a high particle density (i.e., Tc —2b-C2i)) but still assuming a uniform N (see Section 5.3), the combination of Eqs. (5.12) and (5.13) yields that log(cr) varies as rather than the above log(cr) oc result of the dilute Nlimit [11] that is given by Eq. (5.14). This yields that the conductance of the system, as a function of x, will be given by [ 11 ]... 

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]. 

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 theory is commonly applied to describe phase transitions or transport phenomena in random heterogeneous materials. These kinds of critical processes are characterized by strong divergence of certain physical quantities in the near of the so-called percolation threshold. The percolation threshold is a critical amount or probability for the appearance of variously shaped heterogeneities (denoted by subscript c), characterized by a strong increase of the connectivity or the formation of endless clusters. 

For the polymers containing filler that touch each other, the percolation theory has been developed. This assumes a sharp increase in the effective conductivity of the disordered media, polymer matrix composite, at a critical volume fraction of the reinforcement known as the percolation threshold (( )percoi) which long-range connectivity of the system appears. The model that best expresses these aspects is the one created by Vysotsky (Vysotsky and Roldughin 1999), which presumes a percolation network of nanofiller particles inside the polymer matrix as shown in equation (11.10) ... 

Electrically conductive polymer nanocomposites are widely used especially due to their superior properties and competitive prices. It is expected that as the level of control of the overall morphology and associated properties increases we will see an even wider commercialisation on traditional and totally novel applications. In this section we have discussed the basic principles of the percolation theory and the different types of conduction mechanisms, outlined some of the critical parameters of controlling primarily the electrical performance and we have provided some indications on the effect such conductive fillers have on the overall morphology and crystallisation of the nanocomposite. The latter becomes even more critical if we take into consideration that modem nanosized fillers offer unique potential for superior properties at low loadings (low percolation thresholds) but have a more direct impact on the morphology of the system. Furthermore we have indicated that similar systems can have totally different behaviour as the preparation methods, the chain conformation and the surface chemistry of the fillers will have a massive... 

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