Percolation theory solids

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 conductivity of the alloys as a function of nanotube concentration is shown in Fig. 5.9. P30T is an insulator with a conductivity 10-8 S/m. The conductivity increases monotonically with nanotube concentration. However the rate of increase is relatively small at low concentrations. It increases dramatically above 12% concentration. Above this concentration the nanotubes are close enough so that percolation becomes possible. The fit of the percolation theory is shown by the solid line. 

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] ... 

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. 

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. 

As mentioned before, the disordered solids will be mostly modelled in this book using randomly diluted site or bond lattice models. A knowledge of percolation cluster statistics will therefore be necessary and widely employed. Although this lattice percolation kind of disorder will not be the only kind of disorder used to model such solids, as can be seen later in this book, the widely established results for percolation statistics have been employed successsfully to understand and formulate analytically various breakdown properties of disordered solids. We therefore give here a very brief introduction to the percolation theory. For details, see the book by Stauffer and Aharony (1992). 

As discussed earlier, the percolation theory or similar theories of the cluster statistics in disordered solids can give us the probability density g l) of the defect clusters of linear size 1 ... 

Application of Percolation Theory to Describing Kinetic Processes in Porous Solids... 

Most of the pore structures (e.g., spongy structures) consist of extensive three-dimensional networks in which there is a profusion of interconnections between voids within the structure. The latter interconnections affect considerably the kinetics of various processes in porous solids. This effect can adequately be described by employing the ideas developed in percolation theory 7-13). In the framework of this theory, the medium is defined as an infinite set of sites interconnected by bonds. Percolation theory can be applied to porous solids via identification of network sites with voids, and bonds with necks. Thus, the theory is applicable primarily to spongy porous structures but in some cases also to corpuscular structures. 

During the past decade, percolation theory has been successfully used to analyze condensate desorption from porous solids (14-34), mercury penetration into porous solids (35 -43), and the kinetics of catalytic deactivation... 

In the following discussion we will consider the application of percolation theory to describing desorption of condensate from porous solids. In Sections III,A-III,C we briefly recall types of adsorption isotherms, types of hysteresis loops, and the Kelvin equation. The matter presented in these sections is treated in more detail in any textbook on adsorption [see, e.g., the excellent monographs written by Gregg and Sing (6) and by Lowell and Shields (49) Sections III,D-III,H are directly connected with percolation theory. In particular, general equations interpreting the hysteresis loop are... 

Application of percolation theory to describing the desorption process from porous solids is based on the identification of network sites with voids, and bonds with necks. A bond is considered to be unblocked if the neck radius r > Kp. Unblocked sites belonging to the percolation cluster correspond to voids containing nitrogen vapor. 

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... 

Analyzing adsorption (Section III,C) and desorption (Sections ni,D and III,E), we assumed that the pore volume is concentrated in voids, whereas necks do not possess volumes of their own (Fig. 2). Seaton (34) has recently considered an alternative model of porous solids assuming the pore volume to be concentrated in necks (Fig. 3). In the framework of this model, the adsorption process is described by analogy with Eq. (18) (one should only replace the void radius distribution by the neck radius distribution), and consequently the analysis of the adsorption branch of the isotherm allows one to obtain the neck-size distribution. The desorption process can be described by using the same ideas as in Sections III,D and III,E because this process is mathematically equivalent to the bond problem in percolation theory, even if the pore volume is concentrated in the necks. In particular, the volume fraction of emptied necks under desorption [1 - C/des(fp)] can... 

We now consider application of percolation theory to describing mercury intrusion into porous solids. First we briefly recall the main physical principles of mercury porosimetry (in particular, the Washburn equation). These principles are treated in detail in many textbooks [e.g., Lowell and Shields 49)]. The following discussions (Sections IV,B and IV,C) introduce general equations describing mercury penetration and demonstrate the effect of various factors characterizing the pore structure on this process. Mercury extrusion from porous solids is briefly discussed in Section IV,D. 

During mercury intrusion, a given void or neck with r > rp can be filled by mercury only if it is connected with the outer surface by a chain of voids and necks with r > rp. Thus, mercury intrusion into porous solids is equivalent to the bond problem in percolation theory [Androutsopoulos and Mann (35), Wall and Brown (14), Chatzis and Dullien (36), Lane et al. (37), Zhdanov and Fenelonov (38), Tsakiroglou and P atakes (39-41), Day et al. (42), and Park and Ihm (43)]. The equivalence is based on the identification of network sites with voids, and bonds with necks. A bond is considered to be unblocked if the neck radius r > rp. 

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