Percolation theory approach

Yanuka, M. 1992. Percolation theory approach to transport phenomena in porous media. Transp. Por. Media 7 265-282. 

It was shown that it is fruitful to use the percolation theory approach for studying properties of phases with wide HR. 

The common disadvantage of both the free volume and configuration entropy models is their quasi-thermodynamic approach. The ion transport is better described on a microscopic level in terms of ion size, charge, and interactions with other ions and the host matrix. This makes a basis of the percolation theory, which describes formally the ion conductor as a random mixture of conductive islands (concentration c) interconnected by an essentially non-conductive matrix. (The mentioned formalism is applicable not only for ion conductors, but also for any insulator/conductor mixtures.)... 

Alternatively, Leung and Eichinger [51] proposed a computer simulation approach which does not assume any lattice as the classical and percolation theory. Their simulations are more realistic than lattice percolation, since spatially closer groups form bonds first and more distant groups at later stages of network formation. However, the implicitly introduced diffusion control is somewhat obscure. The effects of intramolecular reactions were more realistically quantified, and the results agree quite well with experimental observations [52,53],... 

Hsu and Berzins used effective medium theories to model transport and elastic properties of these ionomers, with a view toward their composite nature, and compared this approach to that of percolation theory. ... 

As discussed above, hysteresis loops can appear in sorption isotherms as result of different adsorption and desorption mechanisms arising in single pores. A porous material is usually built up of interconnected pores of irregular size and geometry. Even if the adsorption mechanism is reversible, hysteresis can still occur because of network effects which are now widely accepted as being a percolation problem [21, 81] associated with specific pore connectivities. Percolation theory for the description of connectivity-related phenomena was first introduced by Broad-bent et al. [88]. Following this approach, Seaton [89] has proposed a method for the determination of connectivity parameters from nitrogen sorption measurements. 

The theoretical model describes the break up of the coal macromolecular network under the influence of bond cleavage and crosslinking reactions using a Monte Carlo statistical approach (32-38). A similar statistical approach for coal decomposition using percolation theory has been presented by Grant et al. (39). 

Various mechanisms of coke poisoning active site coverage, pore filling as well as pore blockage have been observed in FCC [18, 19, 43] and Percolation theory concepts have been proposed for the modelling here of [45, 46, 47, 48]. This approach provides a framework for describing diffusion and accessibility properties of randomly disordered structures. 

Some experimental studies point out that the diffusion rate of pure hydrocarbons decreases with the coke content in the zeolite [6-7]. Theoretical approaches by the percolation theory simulate the accessibility of active sites, and the deactivation as a function of time on stream [8], or coke content [9], for different pore networks. The percolation concepts allow one to take into account the change in the zeolite porous structure by coke. Nevertheless, the kinetics of coke deposition and a good representation of the pore network are required for the development of these models. The knowledge of zeolite structure is not easily acquired for an equilibrium catalyst which contains impurity and structural defects. 

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. 

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 ... 

Computer modelling of physisorption hysteresis is simplified if it is assumed that pore filling occurs reversibly (i.e. in accordance with the Kelvin equation) along the adsorption branch of the loop. Percolation theory has been applied by Mason (1988), Seaton (1991), Liu et al., (1993, 1994), Lopez-Ramon et al., (1997) and others (Zhdanov et al.,1987 Neimark 1991). One approach is to picture the pore space as a three-dimensional network (or lattice) of cavities and necks. If the total neck volume is relatively small, the location of the adsorption branch should be mainly determined by the cavity size distribution. On the other hand, if the evaporation process is controlled by percolation, the location of the desorption branch is determined by the network coordination number and neck size distribution. 

Leuenberger, H., Rohera, B. D., and Haas, C. (1987), Percolation theory—A novel approach to solid dosage form design, Int. J. Pharm., 38,109-115. 

Holman, L. E., and Leuenberger, H. (1988), The relationship between solid fraction and mechanical properties of compacts—The percolation theory model approach, Int. J. Pharm., 46, 35-44. 

Basically, the process of tablet compression starts with the rearrangement of particles within the die cavity and initial elimination of voids. As tablet formulation is a multicomponent system, its ability to form a good compact is dictated by the compressibility and compactibility characteristics of each component. Compressibility of a powder is defined as its ability to decrease in volume under pressure, and compactibility is the ability of the powdered material to be compressed into a tablet of specific tensile strength [1,2], One emerging approach to understand the mechanism of powder consolidation and compression is known as percolation theory. In a simple way, the process of compaction can be considered a combination of site and bond percolation phenomena [5]. Percolation theory is based on the formation of clusters and the existence of a site or bond percolation phenomenon. It is possible to apply percolation theory if a system can be sufficiently well described by a lattice in which the spaces are occupied at random or all sites are already occupied and bonds between neighboring sites are formed at random. 

Water is well known for its unusual properties, which are the so-called "anomalies" of the pure liquid, as well as for its special behavior as solvent, such as the hydrophobic hydration effects. During the past few years, a wealth of new insights into the origin of these features has been obtained by various experimental approaches and from computer simulation studies. In this review, we discuss points of special interest in the current water research. These points comprise the unusual properties of supercooled water, including the occurrence of liquid-liquid phase transitions, the related structural changes, and the onset of the unusual temperature dependence of the dynamics of the water molecules. The problem of the hydrogen-bond network in the pure liquid, in aqueous mixtures and in solutions, can be approached by percolation theory. The properties of ionic and hydrophobic solvation are discussed in detail. 

Mason (18-20) and Palar and Yortsos (26,27) have employed another way of describing desorption from porous solids. Their approach is based on the assumption that the neck arrangement is random, i.e., the probability for an arbitrary neck to have a given value of the radius does not depend on the sizes of adjacent voids and necks. In this case, one can apply the percolation theory data obtained for the bond problem to all the voids. In particular, the probability for an arbitrary void to be empty during the desorption process is precisely 9 b(zo ), where the parameter z is given by Eq. (23). The latter probability is calculated for all the voids. We, however, know for a fact that voids with r < rp are filled. Thus the probability for a void with r > rp to be empty is just 9, (zoq)/F(rp), where F(rp) is the fraction of voids with r > rp [Eq. (33)]. Then, by analogy with Eq. (20), we derive... 

From the data presented in Fig. 27, we can conclude that the approach based on percolation theory permits one to obtain the neck-size distribution only in a relatively narrow range of radii. This is due to the fact that the percolation probability 9 b(z ) has a threshold (3>b(zq) = 0 at zq < 1.5) and increases from 0 to 1 in a relatively narrow range of 1.5 < z < 2.7. 

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). 

Thus, it appears that relative permeability curves follow percolation theory, since they satisfy both the theoretical percolation threshold and the scaling law for three dimensional networks [9]. More importantly, relative permeability curves of different connectivity exhibit the same behavior with as is approached. The same conclusion is valid for different pore size distribution functionsprovided that f.y< 8 [II],... 

This notion is supported by a large number of independent experimental data, related to structure and mobility in these membranes. It implies furthermore a distinction of proton mobility in various water environments, strongly bound surface water and liquidlike bulk water, and the existence of water-filled pores as network forming elements. Appropriate theoretical treatment of such systems involves random network models of proton conductivity and concepts from percolation theory, and includes hydraulic permeation as a prevailing mechanism of water transport under operation conditions. On the basis of these concepts a consistent approach to membrane performance can be presented. 

Thus the model in Figure 3 is consistent with spectroscopic and diffusional results, but is certainly an oversimplified picture nevertheless. Other approaches to the modeling of transport in Nafion, such as the recent application of percolation theory by Hsu and co-workers (22), may yield further insight into the problem. 

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