Percolation theory/threshold

Figure 2.16. The analogy between (a) the gelation threshold and (b) the percolation-theory threshold. Reprinted from Figure 17 (Stanley, 1985). Reprinted with permission of John Wiley and Sons, Inc. Copyright (1985).

The main notion of the percolation theory is the so-called percolation threshold Cp — minimal concentration of conducting particles C at which a continuous conducting chain of macroscopic length appears in the system. To determine this magnitude the Monte-Carlo method or the calculation of expansion coefficients of Cp by powers of C is used for different lattices in the knots of which the conducting par-... 

Experimental dependences of conductivity cr of the CPCM on conducting filler concentration have, as a rule, the form predicted by the percolation theory (Fig. 2, [24]). With small values of C, a of the composite is close to the conductivity of a pure polymer. In the threshold concentration region when a macroscopic conducting chain appears for the first time, the conductivity of a composite material (CM) drastically rises (resistivity Qv drops sharply) and then slowly increases practically according to the linear law due to an increase in the number of conducting chains. 

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 ... 

Where a is the composite conductivity, a0 a proportionally coefficient, Vfc the percolation threshold and t an exponent that depends on the dimensionality of the system. For high aspect ratio nanofillers the percolation threshold is several orders of magnitude lower than for traditional fillers such as carbon black, and is in fact often lower than predictions using statistical percolation theory, this anomaly being usually attributed to flocculation [24] (Fig. 8.3). 

Wool and Cole (6) described a simulation model based on percolation theory for predicting accessibility of starch in LDPE to microbial attack and acid hydrolysis. This model predicted a percolation threshold at 30% (v/v) starch irrespective of component geometry, but the predicted values are not in accordance with results of enzymatic or microbial attack on these materials (Cole, M.A., unpublished data). Since a model that incorporates component geometry provides a better fit to experimental data than a geometry-independent model does, development of advanced models should be based on material geometry and composition, rather than on composition alone. 

The percolation model, which can be applied to any disordered system, is used for an explanation of the charge transfer in semiconductors with various potential barriers [4, 14]. The percolation threshold is realized when the minimum molar concentration of the other phase is sufficient for the creation of an infinite impurity cluster. The classical percolation model deals with the percolation ways and is not concerned with the lifetime of the carriers. In real systems the lifetime defines the charge transfer distance and maximum value of the possible jumps. Dynamic percolation theory deals with such case. The nonlinear percolation model can be applied when the statistical disorder of the system leads to the dependence of the system s parameters on the electrical field strength. 

A simple consideration of granular metals in the framework of the classic percolation theory when granules are treated as metal balls, embedded into insulating material, appears to be very limited. Taking into account quantum effects and, first of all, possibility of the tunnel transitions between nanogranules leads to the change in parameters of the percolation theory and even to diminishing of the percolation threshold [15,38,39]. Even in... 

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] ... 

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] ... 

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]... 

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 ... 

Leuenberger et al. introduced percolation theory in the pharmaceutical field in 1987 to explain the mechanical properties of compacts and the mechanisms of the formation of a tablet [36,37]. Knowledge of the percolation thresholds of a system results in a clear improvement of the design of controlled-release dosage forms such as inert or hydrophilic matrices. 

Study of Ternary Tablets Percolation theory has been developed for binary systems, however, drug delivery systems usually contain more than two components. The existence and behavior of the percolation thresholds in ternary pharmaceutical dosage forms have been studied [39] employing mixtures of three substances with very different hydrophilicity and aqueous solubility (Polyvinylpyrrolidone (PVP) cross-linked, Eudragit RS-PM, and potassium chloride). 

In addition, when the obtained drug percolation thresholds were plotted as a function of the drug-excipient particle size ratio of the matrices (see Figure 25), a linear relationship was found between the drug percolation threshold and the relative drug particle size [46]. These results are in agreement with the above exposed theoretical model based on percolation theory. 

In 1991, Bonny and Leuenberger [40] explained the changes in dissolution kinetics of a matrix controlled-release system over the whole range of drug loadings on the basis of percolation theory. For this purpose, the tablet was considered a disordered system whose particles are distributed at random. These authors derived a model for the estimation of the drug percolation thresholds from the diffusion behavior. 

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. 

The principles of the percolation theory were applied to design controlled-release matrix tablets containing acyclovir in order to estimate the percolation threshold of the excipient in acyclovir matrix tablets and to characterize the release behavior of these hydrophilic matrices in order to rationalize the design of these controlled-release systems. 

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. 

Percolation theory is helpful for analyzing disorder-induced M-NM transitions (recall the classical percolation model that was used to describe grain-boundary transport phenomena in Chapter 2). In this model, the M-NM transition corresponds to the percolation threshold. Perhaps the most important result comes from the very influential work by Abrahams (Abrahams et al., 1979), based on scaling arguments from quantum percolation theory. This is the prediction that no percolation occurs in a one-dimensional or two-dimensional system with nonzero disorder concentration at 0 K in the absence of a magnetic field. It has been confirmed in a mathematically rigorous way that all states will be localized in the case of disordered one-dimensional transport systems (i.e. chain structures). 

---